In the beginning, you know that you have a 1/3 chance of choosing the right door. You might think that, after one of the options is eliminated, now your door and the remaining door both have an equal 1/2 chance to be hiding the prize. This would seem reasonable, as probability suggests that distribution of chance should be even across your options. However, this is not the case, because information is in fact given regarding the choices. By switching doors, you actually raise your chances of winning from 1/3 to 2/3. Here is an explanation.
In the beginning, each door has an equal chance of hiding the prize. You choose a door, and it has 1/3 chance of being the right one. Let's say you are going to stay with this door. At this point, we don't really care about the other doors anymore, because our choice remains static regardless of any information given. Because we're staying with this door, and if it's the right door, we will assuredly win, then we have 1/3 chance of winning. Therefore staying with the door you first chose has effectively a 1/3 chance of winning.
Let's say you were in fact wrong in your original door choice. Then what happens is that the show's host eliminates the other wrong choice for you (because he can't reveal the winning door). At this point, you will win the prize if you switch. Now, since there was a 2/3 chance of being wrong on the first choice, this means you have a 2/3 chance of winning if you switch!
With that, you can see how math will help you if you are ever on a game show!
Still not convinced? Try the Monty Hall game out yourself, below. Or, if that's too slow, you can use the automated game system below the statistics for faster results.