I think part of it is that the answer depends on the fact that the host's strategy is to always open a door with a goat behind it that the player didn't choose.

I think part of it is that the answer depends on the fact that the host's strategy is to always open a door with a goat behind it that the player didn't choose. If the host instead opened one of the doors the player didn't choose at random, then one third of the time the prize will be revealed. The remaining two thirds of the time, switching and keeping your choice have an equal probability of getting you the prize. Many descriptions of the problem don't tell you what the host's strategy actually is, so I'd guess that a lot of people who are "fooled" by the problem actually just have a different reasonable interpretation of what it's asking.