[uchetrip]

The simplest why I have found to explain it to people who don't study conditional probabilities is this (it's pretty much a description of a probability tree):

1) You make an initial choice of 1 in 3, each with equal probability.
2) You are then shown that one of the doors has nothing behind it.
3) The car is now behind one door, giving you a 50/50 chance selection of change or stick, but you already made a 1 in 3 decision, so if you stick with either door you have a 1 in 6 chance, or a total probability of 1 in 3 of winning the car (two lots of 1 in 6 for each door taking all of the remaining initial choices into account)
4) Your change winning probability is therefore 1 - 1/3 = 2 in 3.

People often think you can discount all of the external probabilities that don't affect you, but you can't.