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How does us checking the flop and betting the flop magically change the cards he holds?
It doesn't change the cards he actually holds, but it widens his range. Here's a couple of examples.
Let's say that before the flop we estimate his range to be Ax, 22+ any suited connector and any 2 cards T and higher. Some hands are less likely because he didn't reraise (for example KK or AK) but not impossible.
How does this range change when we bet the flop and he calls? Obviously he is folding hands that have no chance like suited connectors. Many players will also fold hands that AK now beats, like 55-JJ except of course those hands that hit the jackpot. These are the hands we don't want to fold. If we can give them reason to believe that we don't have AK and their 88 might still be good, they can pay us off. By betting the flop we are encouraging these hands to fold. In other words, by betting the flop we're helping worse hands to play perfectly against us.
If we check the flop we're helping that same range of hands to make a mistake. Once we check the flop his range on the turn is exactly what it was before the flop, only now he is more likely to think our range doesn't include AK. Against the marginal hands like 88, 76, etc we can often get 2 streets of value by betting both the turn & river. If he bests, we obviously call both the turn and usually the river.