I'm copying my reply to xrosswind here too, but I'm not sure if it adds any value. At least someone can point to any mistakes I made in the comment.
Hero open raised Qh 7d from button, SB fold, BB three bets and hero calls.
Flop is Kd 7h 3c
BB bets and hero calls. Hero then says if the K had been something like a 9 then he would have raised. Oink can you explain why this is the case.
I think one argument for calling is the simple fact that the villain is much stronger here equity-wise. Also, having 3-bet the flop, he's unlikely to give up just yet with hands like 88-QQ, especially the stronger of those, TT-QQ. Any King will certainly 3-bet the flop right away. Some people will argue that it will be "cheap to get out of the hand" if the villain then 3-bets, but this line doesn't give villain the opportunity to bluff. (First, 3-bet bluffing here after a flop raise would require a pretty aggressive villain, but it would be pretty much impossible for hero to continue UI which means that hero doesn't gain much if the villain 3-bet bluffs.) In this spot, he's much more likely to continue firing on the turn with almost his full range, because he is out of position and he has 3-bet preflop.
Also, I don't remember if Oink mentioned it, but with solid reads, you can probably just call twice and fold to a third barrel. Many TAGs will probably give up on the river once you've called twice on a board like this, so it's unlikely that your pair of sevens is any good after that. It really looks like you're planning to go to the showdown. So, in a sense, the K on the flop keeps the villain more honest on the river especially, but is still likely to fire a second one on the turn. This "call twice and give up on the river" having too few outs to improve, looks kinda silly in hindsight, but it's really difficult for the player in BB's shoes to keep firing on a board like this, OOP, against a player who clearly is going to showdonw.
And as you notice, the actual hand demonstrated the power of Oink's line, because the villain did just what Oink wanted! :-)
But I think there's merit to getting him to fold hands like AQ-A whatever-he's-3-betting-preflop-with on the flop. It's 9,5:1 to call a flop raise for the villain, so it's technically a mistake for him to fold a hand with six live outs. And I think a turn bet will sometimes fold hands like 88-99 maybe even TT, depening on the card of course. (Ace on the turn would be a small disaster for hero.) All this makes me think that getting the villain fold right away on the flop isn't that bad a result.