Monty Hall from the host's POV
The game: there are three doors, one hiding a good reward and two hiding a bad reward. The contestant chooses a door but does not open it. My job is to open one of the other doors, making sure to choose one which hides a bad reward. Then I offer the contestant the choice to switch their choice to the other, unchosen, unopened door. Should they?
If the contestant originally picked the “good” door, then I can open either of the other ones. No matter which I open, if they switch, they’ll get it wrong. So if their strategy is to switch, 1⁄3 they pick the good door originally and then lose by switching to the remaining bad door.
If they picked a bad door originally, my job is easy– I don’t have a choice, I just open the other bad door. Then if they switch, they are switching to the good door. So under the switching strategy, 2⁄3 of the time they pick a bad door and then switch to the good door.
So the switching strategy leads to 2⁄3 chances of success, up from the 1⁄3 chances of the don’t-switch strategy.
From my perspective, it seems clear why it’s even possible for their chances to increase: I look at which door hides what and in the situation where they picked a bad door, I basically flat out tell them which door to choose (by opening the only other bad door). Of course, they don’t know when they picked a bad door originally, but it’s most of the time, so they are well served by taking my hint.