The difficult part of this puzzle is understanding that there is a difference (however subtle) between the door you picked and the door Monty leaves closed. I heard it best described (without mathematics) like so:

The key is to realize that Monty has certain information (i.e. the car's location) and that by purposefully avoiding one of the doors, he has given you new information.

Say that you choose door A, and he reveals door B to be empty.
Before he reveals the door, you know only one thing:
1) The car is behind one of the doors, A, B, or C.
When he reveals door B, you have been given two new pieces of information:
2) B is empty
3) Monty *chose* to open B and to avoid opening C.

Because of that last fact, you can make a better informed choice, which is to switch to C.

Of course, it may be more convincing just to try the game at home or write out the probability tree.