Say, for example, that you have a bunch of farmers growing crops. Eventually, some of these people get different jobs instead: helping sick people, developing new farming tools, maintaining these tools, and even invading neighbouring groups of people. These people have become a civilization, although what with all the invading, it's debatable how civilized they are. Now only the farmers have the time to grow their own food, but everyone else still needs to eat. The civilization needs the farmers to grow more crops than they can personally eat, in order to support everyone else. Money and taxes are invented. The population increases, more different jobs come into existence, fewer people learn how to farm, and the people run out of neighbours to invade. This causes increasing strain to be placed on the farmers. There's less food to go around. Inflation of the money occurs. Taxes go up. No one can afford to feed themselves. The people either give up on their civilization, or are invaded by a more recent neighbouring civilization that's still on the rise.

Which explains why previous civilizations, such as the Roman Empire, collapsed (unless you share the same beliefs as Philip K. Dick). It might also explain why I always lose at SimCity.

As far as I can tell, the only way to avoid this is to encourage everyone to have less children, keeping the population roughly constant; to somehow ensure that no one will be unemployed or otherwise contributing nothing important to society; and to not rely on any external factors that will eventually run out, such as the constant invasion of neighbouring people or the extraction of energy from a limited supply of fossil fuels.

See also: the law of diminishing returns, the iron law of wages, and Malthusian catastrophe.

Just to clarify, I'm not suggesting that it's impossible for a civilization to exist indefinitely, only that it's impossible for it to increase its population indefinitely. Although technology allows more people to live on any given size of land, it won't allow the population to continue increasing forever.