When you start there are 3 doors and you pick one. The chance of the car being behind that door is 1/3. The chance of it being behind either of the other two doors is 2/3.

When Monty opens one of the other doors these probabilities do not change, but one of the other doors has been eliminated (shows a goat).

Hence the single other door left has the probability of 2/3 and is the one you should choose.

In opening the door and showing the goat Monty has communicated a piece of information to you.

BUT.

What if Monty doesn't know which door has the car?

It starts off the same with the probability of the first door you pick having the car being 1/3 and of either of the other doors having the car as 2/3.

When Monty opens a door then it will either be the car (which you automatically win) or a goat.

The chances of you winning the car automatically in this way is 1/2 x 2/3 = 1/3

But what if Monty's choice shows the goat? Is it still better to switch?

No, in this case the final two doors each have probability 1/3 of having the car and switching makes no difference. So although the physical events occurring are the same as the first problem the result is different, just because of a different state in Monty's brain!

The usual manner in which the story is related does not make clear that Monty is acting with knowledge about where the car is and so it can be argued that peoples usual response is not entirely wrong.