I don't think he leaks information. He invites you to switch from a game with a 1-in-3 chance to a game with a 2-in-3 chance. If you stick, you are staying in the 1-in-3 game, but if you switch then you are changing to a 2-in-3 game. That's all there is to it.

Edit: I should have checked before I posted, obviously it's 2-in-3 when you switch. But it's same idea, you have better odds when you switch because then it's like you are entering a new game with only 2 doors.

Still hard to believe that Erdos wouldn't believe this. He spent his life couch surfing at other mathematicians but not could convince him until he saw a monte-carlo simulation, apparently.