I think the intuition behind is that, when you initially choose one door out of 1000000, you are almost sure not to have picked the right one, so you ignore the fact that you even chose one. Then, the host opens 999998 doors, but leaves one closed. That door is now really special!
Or a different way to look at it: The host gives you partial information about what's between the 999999 doors you didn't choose. Doesn't it seem really unlikely that your intial choice matters given that there are so many doors?
The problem to me is more that when you go back down from 1M to 3 doors one can just as well say again "but now we only have three doors". Meaning that a proof of the original problem is the only satisfactory answer; the 1M comparison is maybe for people that don't like proofs and want "evidence".
This explanation offered above makes sense to my "gut" however: https://news.ycombinator.com/item?id=21349295