Imagine a similar but slightly different scenario: We now have two contestants: Mr A and Mr B. Mr A is the contestant in the original story. He gets to pick a door. The host then opens a door with a goat (and removes it from the stage). Then Mr A gets to choose whether to keep his door or open the third.

Now the second contestant, Mr B, joins the show. He has not seen anything prior and has no knowledge of Mr A or the other door previously removed from stage. All he sees is two doors. He has the option of opening one of the two.

Mr A has 2/3 chance of switching door and coming out with a car. Whereas Mr B has 1/2 chance (from his perspective) of picking either door and coming out with a car.

I think the main problem why so many people have a hard time grasping the solution is that they're looking at the probability as Mr B and ignoring the extra information they do have as Mr A.