The most convincing and intuitively palatable approach to this problem IMO is to make diagrams of your choices and the possible outcomes. While one may argue the philosophical point that nothing has changed between the first choice and the first reveal, making a diagram of each strategy makes it clear that a strategy of switching after the reveal will result in a successful outcome (in this case, apparently, some interaction with the comley Ms. White) two-thirds of the time vs sticking with the original choice.
Short of the diagrams, here is my best explanation™ of why switching works: When you first picked, you had a 1/3 chance of picking the winner and a 2/3 chance of picking a loser. Once shown one of the losing options, the strategy of switching will only result in a bad outcome in the event that you had picked the right one in the first place. And since you originally had only a 1/3 chance of winning, you now have only a 1/3 chance of losing if you make the switch, vs a 2/3 chance of winning.
