In the world of simple exchange, money is just a convenience for enabling the exchange of commodities: C-M-C is easier to arrange than C-C. But profit-making business is different: the sequence there is M-C-M'. The capitalist enters the market and buys some commodities for a certain sum of money. Later, he sells some commodities, and has a larger sum of money. This increase -- from M to M' -- is the whole point of being capitalist. But in a world of free market exchange, how can it exist?
Let's put some obvious misunderstandings out of the way. There's nothing mysterious about the fact that people can accomplish things with tools and previously acquired materials that they would be unable to with unaided labor. The problem is not that "capital," in the sense of a stock of tools and materials, is productive in this sense. To the extent that what appears as "profit" in the national accounts is just the cost of replacing worn-out tools and materials, there's no puzzle. 
The mystery is, how can someone enter the market with money and, after some series of exchanges, exit with more money? In the sequence M-C-M', how can M' be greater than M? How can the mere possession of money seemingly allow one to acquire more money, seemingly without end?
Before trying to understand Marx's answer, let's consider how non-Marxist economists answer this question.
1. Truck and barter. The most popular answer, among both classical and modern economists, is that the M-C-M' sequence does not exist. All economic activity is aimed at consumption, market exchange is only intended to acquire specific use-values; when you think you see M-C-M you're really looking at part of some C-M-C sequence(s). The classical economists are full of blunt statements that the only possible end of exchange is consumption. In today's economics we find this assumption in the form of the "transversality condition" that says that wealth must go to zero as time goes to infinity. That's right, it is an axiom in modern economics that accumulation cannot be a goal in itself. Or in the words of Simon Wren-Lewis (my new go-to source for the unexamined conventional wisdom of economists): "It would be stupid to accumulate infinite wealth." Well OK then!
2. You earned it. Another answer is that the capitalist brings some additional unmeasured commodity to the production process. They are providing not just money M but also management ability, risk-bearing capacity, etc. In this view, if we correctly measured inputs, we would find that M'=M. In its most blatantly apologetic form this is effectively skewered by comrades Ackerman and Beggs in the current Jacobin. For unincorporated businesses, it is true, it is not straightforward to distinguish between profits proper and the wages of managerial labor, but that can't account for profits in general, or for the skewed distribution of income across households. If anything, much of what is reported as managerial salaries should probably be called profits. This is a point made in different ways by Piketty and Saez and Dumenil and Levy; you can also find it offered as straightforward business advice.
3. It was the pictures that got small. The other main classical answer is that profit is the reward for "abstinence" (Senior) or "waiting" (Cassel). (I guess this is also the theory of Bohm-Bawerk and the other Austrians, but I admit I don't know much about that stuff.) It appears today as a discount rate on future consumption. This invites the same question as the first answer: Is capitalist accumulation really motivated by future consumption? It also invites a second question: In what sense is a good tomorrow less valuable than the same good today? Is the utility derived from a glass of wine in 2013 really less than the utility derived from the same glass consumed in 2012,or 2010, or 1995? (So far this has not been my experience.) The logically consistent answer, if you want to defend profit as the return to waiting, is to say Yes. The capital owner's pure time preference then represents an objective inferiority of output at a later date compared with the same output at an earlier date.
This is a logically consistent answer to the profits puzzle, and it could even be true with the right assumptions about the probability of an extinction-event asteroid impact/Khmer Rouge takeover/zombie apocalypse. With a sufficiently high estimate of the probability of some such contingency, M' is really equal to M when discounted appropriately; capitalists aren't really gaining anything when you take into account their odds of being eaten by zombies and/or suffocated by plastic bag, before they get to enjoy their profits.  But I don't think anyone wants to really own this point of view -- to hold it consistently you must believe that economic activity becomes objectively less able to satisfy human needs as time goes by. 
4. Oops, underpaid again. We can take the same "profit as reward for waiting" idea, but instead of seeing a pure time preference as consistent with rational behavior, as modern economists (somehow) do, instead interpret it like the classical economists (including Cassel, whose fascinating Nature and Necessity of Interest I just read), as a psychological or sociological phenomenon. Consumption in the future is objectively identical to the same consumption today, but people for some reason fail to assign it the same subjective value it the same. Either they suffer from a lack of "telescopic facility," or, in Cassel's (and Leijonhufvud's) more sophisticated formulation, the discount rate is a reflection of the human life expectancy: People are not motivated to provide for their descendants beyond their children, and future generations are not around to bargain for themselves. Either way, the outcome is that exchange does not happen at value -- production is systematically organized around a higher valuation of goods today than goods tomorrow, even though their actual capacity to provide for satisfaction of human needs is the same. Which implies that workers -- who provide labor today for a good tomorrow -- are systematically underpaid.
5. Property is theft. The last and simplest possibility is that profits are always just rents. Capitalists and workers start out as just "agents" with their respective "endowments." By whatever accident of circumstances, the former just end up underpaying the latter. Maybe they are better informed.
We could develop all these points further -- and will, I hope, in the future. But I want to move on to (my idea of) Marx's answer to the puzzle.
One other thing to clear up first: profit versus interest. Both refer to money tomorrow you receive by virtue of possessing money today. The difference is that in the case of profit, you must purchase and sell commodities in between. What is the relationship between these two forms of income? For someone like Cassel, interest has priority; profit is a derived form combining interest with income from managerial skill and/or a rent. For Marx on the other hand, and also for Smith, Ricardo, etc., profit is the primitive and interest is the derived form; interest is redistribution of profits already earned in production. (Smith: "The interest of money is always a derivative revenue, which, if it is not paid from the profit which is made by the use of the money, must be paid from some other source of revenue.") In other words, are profits an addition to interest, or is interest as a subtraction from profit? For Marx, the latter. The fundamental question is how money profits can arise through exchange of commodities. 
Marx gives his answer in chapter four: The capitalist purchases labor-power at its value, but gets the results of the labor expended by that labor-power. The latter exceeds the former. In other words, people are capable of producing more than it takes to reproduce themselves, and that increment is captured by the capitalist. In four hours, you can produce what you need to live on. The next four or six or eight or twelve hours, you are working for The Man.
This is the answer, as Marx gives it. Labor power is paid for at its value. But having purchased labor power, the capitalist now has access to living labor, which can produce more than the the cost of its own reproduction.
I think this is right. But it's not really a satisfactory answer, is it? It's formally correct. But what does it mean?
One way of fleshing it out is to ask: Why is it even possible that labor can produce more than the reproduction-costs of labor power? Think of Ricardo's world. Profits are positive because we have not yet reached the steady state -- there are still natural resources available whose more intensive use will yield a surplus beyond the cost of the labor and capital required to use them. The capitalist captures that surplus because capital has the short side of both markets -- there is currently excess land going unutilized, and excess labor going unutilized. 
Another way: There is something in the production process other than exchange, but which is captured via exchange.
I want to think of it this way: Humanity does have the ability to increase social value of output, or in other words the aggregate capacity to satisfy human needs from nature does in general grow over time. This "growth" happens through our collective creative interchange with nature -- it is about pushing into the unknown, a process of discovery -- it is not captured beforehand in the market values of commodities.
In a proper market, you cannot exchange a good in your possession for a good with a greater value, that is, with a greater capacity to satisfy human needs in general. (Your own particular needs, yes.) But you can, through creative activity, through a development of your own potential, increase the general level of satisfaction of human needs. The capitalist by buying labor power at its value, is able to capture this creative increment and call it their private property.
Our potential is realized through a creative interchange with nature. It's not known in advance. What can we do, what can't we do -- we only learn by trying. We push against the world, and discover how the world pushes back, in so doing understand it better and find how it can be reshaped to better suit our needs. Individually or collectively, it's a process of active discovery.
You as a person can exchange the various things you are in possession of, including your labor power, for other things of equal value. (Though for different use values, which are more desired by you.) But you will also discover, through a process of active learning and struggle, what you are capable of, what are the limits of your powers, what creative work you can do that you cannot fully conceive of now.
Through the process of education, you don't just acquire something that you understood clearly at the outset. You transform yourself and learn things you didn't even know you didn't know. When you do creative work you don't know what the finished product will be until you've finished it. I still -- and I hope for the rest of my life -- find myself reading economics and having those aha moments where you say, "oh that's what this debate is all about, I never got it before!" And science and technology above all involve the discovery of new possibilities through a process of active pushing against the limits of our knowledge of the world.
The results of these active process of self-development and exploration form use-values, but they are not commodities. They were not produced for exchange. They were not even known of before they came into being. But while they are not themselves commodities, they are attached to commodities, they cannot be realized except through existing commodities. I may produce in myself, through this process of self-testing, a capacity for musical performance, let's say. But I cannot realize this capacity without, at least, a sufficient claim on my own time, and probably also concrete use-values in the form of an instrument, an appropriate performance space, etc., and also some claim on the time of others. In this case one can imagine acquiring these things individually, but many -- increasingly over time -- processes of self-discovery are inherently collective. Science and technology especially. So specifically a discovery that allows cheaper production of an existing commodity, or the creation of a new commodity with new use-values, can only become become concrete in the hands of those who control the process of production of commodities. By purchasing labor power -- in the market, at its value -- capitalists gain control of the production process. They are thus able to claim the fruits of humanity's collective self-discovery and interchange with nature as their own private property.
In some cases, this is quite literal. Recall Smith's argument that one of the great advantages of the division of labor is that it allows specialized workers to discover improved ways of carrying out their tasks. "A great part of the machines made use of in those manufactures in which labour is most subdivided, were originally the inventions of common workmen, who, being each of them employed in some very simple operation, naturally turned their thoughts towards finding out easier and readier methods of performing it." Who do you think gained the surplus from these inventions? This still happens. Read any good account of work under capitalism, like Barbara Garson's classic books All the Livelong Day and The Electronic Sweatshop. You'll find people actively struggling to do their jobs better -- the customer service representative who wants to get the caller to the person who can actually solve their problem, the bookshelf installer who wants it to fit in the room just right. The results of these struggles are realized as profits for their employers. But these are exceptional. The normal case today is the large-scale collective process of discovery, which is then privately appropriated. Every new technology draws on a vast history of publicly-available scientific work -- sometimes we see this directly as with biomedical research, but even when it's not so obvious it's still there. Every Hollywood movie draws on a vast collective project of storytelling, a general collective effort to imbue certain symbols with meaning. Again see this most directly in the movies that draw on folktales and other public-domain work, but it's true generically.
It is this vast collective effort at transformation of nature and ourselves that allows the value of output to be greater than the value of what existed before it. Without it, we would eventually reach the classical steady state where the exercise of labor could produce no more than the value of the labor power that yielded it. So when Marx says the source of profits is the fact that labor can produce more than the value of labor power, lying behind this is the fact that, due to humanity's collective creative efforts, we are continuing to find new ways to shape the world to our use.
Capital is coordination before it is tangible means of production. Initially (logically and historically) the capitalist simply occupies a strategic point in exchange between independent producers thanks to the possession of liquid wealth; but as the extension of the division of labor requires more detailed coordination between the separate producers, the capitalist takes over a more direct role in managing production itself. "That a capitalist should command on the field of production, is now as indispensable as that a general should command on the field of battle."
There is another way of looking at this: in terms of the extension of cooperation and the division of labor, which is realized in and through capitalist production, but in principle is independent of it. I'll take this up in a following post.
 Marx makes this point clearly in his critique of the Gotha program. Elimination of surplus as such cannot be a goal of socialism.
 It would seem that we have enough evidence to rule out a sufficiently high probability of world-ending catastrophe to explain observed interest rates, assuming the minimum possible return on accumulated wealth is zero. But of course in some conceivable circumstances it could be negative -- that's why I include the Khmer Rouge takeover, where your chance of summary execution is presumably positively related to your accumulated wealth. Also, maybe we have reason to think that catastrophe is more likely in the future than we would naively infer from the past. It would be funny if someone tried to explain interest rates in terms of the doomsday argument.
 There has been a lot of discussion of appropriate social discount rates in the context of climate change. But nobody in that debate, as far as I can tell, takes the logical next step of arguing that excessively high discount rates imply a comprehensive market failure, not just with respect to climate change. There is not a special social discount rate for climate, there is an appropriate social discount rate for all future costs and benefits. If market interest rates are not the right tool for weighing current costs against future benefits for climate, they are not the right guide for anything, including the market activities where they currently govern.
 Yes, interest exists independently of profits from production, and indeed is much older. Marx recognizes this. But capitalism is not generalized usury.
 And substitution between factors is impossible -- Marx's "iron law of proportions" -- or at least limited.