How does Coriolanus's downfall come about?

Coriolanus's downfall comes about largely because he's not prepared to play the game of democracy. This is a man who's been brought up to despise the mob and to see them as fickle, ignorant, and stupid. The very idea of pandering to the great unwashed is anathema to this proud, snobbish aristocrat and general. Besides, Coriolanus thinks he's done enough for the people of Rome already by bringing them military glory. He doesn't see why he should have to go out of his way to flatter them as well.
This is Coriolanus's biggest problem: he doesn't understand how democracy works. He doesn't understand that the people need to feel a little love from their leaders from time to time. But Coriolanus is neither willing nor able to lower himself to this level. He'd much rather ally with the enemy Volscians than play the game of democratic politics he so heartily detests.
In fact, it's no exaggeration to say that Coriolanus regards the common people of Rome, the plebs, as more of an enemy that the Volscians themselves. This proves to be a fatal miscalculation on the general's part, as he ends up being betrayed and murdered by the leader of the Volscians, Aufidius.