I found the easiest way was this: You pick door A, the host opens door B or C and offers to let you switch to door C or B. He is effectively letting you choose between either A alone (by sticking) or B & C together (by switching) - i.e. by switching you open two doors instead of one, hence twice the probability.

I've found that the most enlightening explanation is to increase the number of doors. What if the game is played with 100 doors, and the host opens 98 of them after you've chosen yours? At the start you only have a 1% chance of winning. Most people see right away that the host can't be increasing the odds to just 50% by opening so many doors. They don't always see the logic immediately, but at least they begin to question their intuition. Getting people to admit that they might be wrong is often the first step toward getting them to accept the correct answer. (That last sentence is a general observation that applies to a much wider range of problems.)