The MHP can be summed up as follows (via Wikipedia:)
Suppose you're on a game show and you're given the choice of three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?
Most people's intuition tells them that it doesn't matter what they do -- both doors have a 50% chance of winning. If you do the math, though, (or run the simulator) it becomes clear that switching gives you a 66.7% of winning the car.
Here is the explanation: when you first pick a door, you have a 33.3% chance of identifying the correct door. But things change when the host has to open a door with a goat. If you have the correct door, which happens 33.3% of the time, he may choose either of the other doors. If you have the wrong one, 66.7% of the time, he has no choice and must pick the only remaining goat-door. In other words, 66.7% of the time, the door he does not pick contains the car.
So how does this apply to religion?
Well, when you are born, you are assigned a random category (a specific religion or atheism or agnosticism or nothing) based on your parents' religion*. This is akin to choosing one of the three doors, because your category was "chosen" out of all existing categories**. However, once you learn a little bit about the world and gain the ability to reason, it's relatively easy to have Monty Hall open some of the false doors -- i.e. you may safely eliminate categories which are clearly incoherent (to you) or simply extremely improbable, like Scientology, Mormonism, and various denominations that hold of Biblical literalism.
You now have a choice akin to the two doors. By the same logic as the MHP, then, and with the added assumption that your religion isn't necessarily more likely to be true than the others***, then your best bet is to switch once you reach the age of knowledge and reason.
Things get more complex if you attempt to rate the categories you can't eliminate according to probability (e.g. you might think Catholicism is twice as likely to be true as Islam, or that Presbyterianism is slightly more likely than Anglicanism) but the overall logic doesn't change. Unless you think you can safely eliminate all other categories or you believe you can fairly rate your own as exceedingly probable, the right move is to switch.
* Obviously, this is an oversimplification, for the sake of clarity.
** The assigned categories are weighted by numbers of adherents and birthrate, of course, but that doesn't necessarily change the underlying logic, since we don't know whether popular religions are more likely to be correct or not.
*** Understanding that people are biased towards (or sometimes against) their given religion should play into this. Also, if you can eliminate your own category, you clearly have to switch.
(See also Bad Religious Arguments: Pascal's Wager, in which I argue that if your primary concern is optimizing your afterlife, you should choose a relatively probable religion that has the worst hell for nonbelievers and the surest way into heaven for adherents.)
EDIT: It occurred to me to google "Monty Hall religion" right after I posted, and it turns out at least one other person had a similar idea.