There's an important variation of the game which should be considered alongside the original to get a better feeling about what is happening here. Suppose that instead of opening the door himself, Monty asks a random person from the audience (who certainly does not know behind which door is the car) to suggest a second door to open after you pick yours.

The door proposed by the person is opened and you see that there is a goat behind it. Hence, besides the fact that a different person now suggested the second door, from your perspective everything happened exactly like it did in the original version.

Should you now change your choice?