Here's an adjustment to the rules of the game that I think will help illustrate why you should always switch.

In my modified game, there are TWO players and no host, and the prize is behind only one of the three doors. Player #1 gets to choose from among the three doors. Player #2 gets to have BOTH of the two doors Player #1 does not pick. In this situation, whose odds are better? Obviously Player #2's odds are better -- he wins 2/3 times while Player #1 only win 1/3. Basically, its better to have Player #2's draw than Player #1's.

Now, return to the original Monty Hall problem where you choose 1 out of 3 doors and the host then eliminates one of the remaining doors. You're effectively Player #1 from my revised example, and your door choice has a 1 in 3 chance of being right. There is a 2 in 3 chance that the prize is behind one of the OTHER two doors (the Player #2 draw). The host is going to eliminate one of those doors in "Player #2's draw" -- but that doesn't change the odds of the #2 draw winning because the host will always eliminate a door that doesn't have a prize. If the prize was behind one of these other two doors, it still will be when a door is eliminated. So when you are offered to switch, if you stay with your original choice, you (like Player #1) win 1 out of 3 times. If you switch to what is effectively Player #2's original draw, you increase your odds to 2 out of 3.

Does that make sense?