It's a cool probability trick.
The sleight-of-hand happens when you assume (as I did, when I first heard it) that the second choice is completely unrelated to the first, and therefore should be calculated fresh as a 50/50 choice. This isn't correct.
The right way to look at it is to start at the beginning. You have a 1/3 chance to pick the prize the first time, and a 2/3 chance to pick a goat. Let's look at each option individually, assuming that we always switch doors at the second stage. If you pick the prize right the first time, and then switch doors, you will obviously fail the game -- this represents a 1/3 chance of failure using our strategy. Now let's say that we picked goat #1 on our initial choice. The host will reveal the second goat, and thus by switching to the third door we win the prize. This is a 1/3 chance of success, based on our strategy. Finally, if we choose goat #2 with our first pick, Monty will then reveal goat #1, ensuring our victory by switching doors. Added with the previous 1/3 chance of success, this makes a 2/3 chance of getting a prize.
So, if you switch doors on the second choice, you have a 2/3 chance at victory -- the only way you fail is if you happen to initially pick the door with the prize. On the other hand, if you stick with your initial choice, you only have a 1/3 chance of success -- you have to nail it the first time, or not at all.