Let's take a moment to consider what that means. ``Capitalism'' is a vague word, but it generally has, as here, two parts, one having to do with markets and one having to do with property. Capitalist economies make extensive use of markets as a means of allocation and distribution, and both production and consumption are oriented towards the market (as opposed to, say, producing purely for one's own personal consumption, or producing for the consumption of one's political master). Capitalist economies also have almost unrestricted rights of private property, including property in land, and tend --- with some notorious exceptions --- to employ free labor (as opposed to slaves or others suffering involuntary servitude). The two components are separable: markets without private ownership have featured in these pages before, as constant readers will recall (no doubt with pain).
Capitalism has often been advanced as one of the reasons why the west (meaning, roughly, western Europeans and their acknowledged descendants in the Americas and Oceania) conquered the world and became the richest and most powerful peoples known to history. The question of whether or not it developed in other parts of the world, which do not have such lurid accomplishments to their name, and why, is thus part of one of the largest and most interesting questions in history and the social sciences.
China is the most obvious rival of Europe for the role of hegemon. (As someone of Indian and Islamic heritage, I say this with a mixture of disappointment and relief; those histories are bloody enough things as they are.) The size of the Chinese cultural and political region has fluctuated over the last fifteen hundred years, but throughout most of that time embraced an area at least as large and populous as western Europe, with (pace European historians disinclined to look at maps) comparable geographic heterogeneity and diversity of resources. Moreover it was typically much more political unified than Europe, so those resources could at least potentially be brought to bear more readily, more advantage taken of that diversity. Why then didn't it take off?
A perfectly sensible answer (given, in somewhat different forms, by both Karl Marx and Max Weber) would lie in political economy. Capitalism drives people to allocate their resources in an instrumentally rational fashion; rams home the habit of instrumental rationality in all spheres of life; and strongly rewards improvements of technique and organization. People differ as to why it developed in Europe (political fragmentation keeping it from being taxed out of existence everywhere at once; bizarre religious delusions; a better system of book-keeping; etc.). It could not develop in China because of the absolutist control exercised by the government, the mandarinate, over Chinese society.
(1) Political suppression. Claiming that the Chinese government was more hostile to initiative than that of, say, Spain or France would take work. (And why should it have been so suppressive?) (2) Scientific backwardness. Traditional Chinese ideas about the natural world were rubbish, of course, but no worse rubbish than those current in Europe before the scientific revolution --- astrology, Aristotle, witchcraft, and all the rest. That revolution didn't even really begin until after 1600; didn't become practically important until the 1700s, by which time Europeans were well on their way to hegemony. (3) Technological backwardness. If anything, China was more technologically advanced than Europe until, again, the 1600s. (4) Capitalism.
History is a sad disappointment to its philosophers and theorists. It would have been a rash alien observer in 1000, or even 1450, who would have bet that Christian western Europeans, of all peoples, would conquer the world and introduce industrialism. They were, after all, still learning how to do anything more complicated than peeling a turnip by copying their neighbors to the south and east, neighbors who were exploiting their technological backwardness and internal divisions to steadily encroach on their territory, bringing a more sophisticated civilization with them.
Still, they did conquer the world, and more or less simultaneously started the scientific and then the industrial revolutions. The consequences were that they and their (acknowledged) descendants became the richest and most powerful peoples in history, forcing essentially all other societies in the world to adopt, at least partially, their ways. Thus it happens that I'm writing this in what should by all rights be the language of a few flea-bitten peasants of a dreary minor island, and not Arabic, Greek, Pali --- or classical Chinese.
China is in some ways the biggest puzzle for this kind of comparative (or: guessing-game) history; it has become a bigger puzzle as we've learned more about the Chinese past.