Here's how I've explained it: Choose randomly between 3 doors. 1/3 of the time you end up with the door with the car, and switching loses. The other 2/3, you pick a door with a goat, the other door with the goat is eliminated, and switching wins.

Basically, P(lose when switching) = P(choosing correct door at first), and P(win when switching) = P(choosing any incorrect door at first).