My intuitive explanation when I was tutoring statistics was the following. (This works better while drawing I find.)
Assume you pick one door. It's the right door with a chance (1/3) and the wrong door with a chance (2/3).
Now if you could pick both other doors (2/3) you would choose those. And this is exactly what you're doing when you switch after the reveal.
The other two doors always have one wrong one, which can be revealed. So switching after the reveal is the same as having the chance to pick two doors before the reveal.
Assume you pick one door. It's the right door with a chance (1/3) and the wrong door with a chance (2/3).
Now if you could pick both other doors (2/3) you would choose those. And this is exactly what you're doing when you switch after the reveal.
The other two doors always have one wrong one, which can be revealed. So switching after the reveal is the same as having the chance to pick two doors before the reveal.