Assume you stick. In that case, you have a 33% chance you've picked the right door, as before. Opening another door doesn't change whether you've picked the right door or not originally.
Assume you switch. In the case you've picked a door with a goat originally (67%), you will necessarily switch to the door with the prize. In the case you've picked a door with the car originally (33%), you will necessarily switch to a door with a goat. Therefore switching gives you a 67% chance of picking the door with the prize.
Another interesting strategy to consider is "randomize" - what if you picked randomly between the two remaining doors? Then you'd have a 50% of getting the prize, since you're selecting randomly between two doors, one of which you know contains the prize.
The decision to switch or not is really a decision between door 1 and door 2. At this point isn't your first guess completely irrelevant?
I've heard this problem and the reasoning before, and for some reason I can't get myself to believe that switching is better.