For more in-depth treatment, see Probabilistic Models of Cognition
Instead of 3 doors, imagine there are 1e9 doors. Then you choose one door (at random), and the host reveals all but one other door.
I think then it's clear why you would switch doors.
What complicates things is that these people are right, too. The generalisation of the problem does not fix the number of doors opened by the host. It could be that the host opens just one door. Or 5e8 doors, splitting the set in the middle. Or any other number in between. In fact, the number of doors opened by the host could be decided by dice before each round starts.
So now the person you talk to goes, "I'm still not convinced, but let's say for the sake of the argument that switching is good in the situation you mention. Now tell me why it is better to switch in all other cases."
You're ending up in formal proof country a lot faster than you'd want for an intuitive explanation.
(Odd observation: if k is the number of doors opened by the host, I find any case where n=2k+1 a lot harder to explain than when k=1 and n is some large number. Maybe that can give some insight into why it's a tough problem.)
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.