For those of you who, like me, couldn't get their head around this problem, try thinking of it this way...
Given a choice between choosing a goat to win a car or choosing the car to win a car, which would you pick?
Choosing the car at the begining of the game gives you a 1 in 3 chance. Choosing a goat gives you a 2 in 3 chance.
Asking to switch in the middle is simply changing this from a "choose the car" game to a "choose the goat" game, as the only way to win by switching is to have picked a goat in the first half.
Therefore, you have a higher probability of winning by switching.