My thought process is in the comments as follows.
~was afraid he had a K on the flop
~bet the pot and he called, certain he had a K
~checked the turn and he checked back so then knew I was ahead
Do you like my reasoning or do you think that I missed anything?
I think your read may have been solid (results-wise, it’s hard to argue that it wasn’t) but I have some caveats/nits to pick:
preflop 3bet was way too small! Don’t use the minraise 3bet/4bet sizing that many players default to on Replay; you are leaving value on the table. Villain might have called all the way up to all-in preflop, and it seems certain they would have at least called a 2.5x reraise.
conversely, IMO flop sizing was too large. Generally, the more often our range wants to bet a given board, the smaller our bet size should be. As the preflop 3bettor we want to cbet almost all K-hi flops to leverage our range advantage; as such, [full pot] is much too large a size. We probably want to bet somewhere between 1/4 and 1/2 pot almost all the time. I could see an argument perhaps for sizing up a bit if you think Villain can have a lot of flush draws in range, meaning we can try to extract more value, but given preflop action I’m not sure how often that applies here.
Having used [full pot] on the flop, we really should have filtered out hands like TT a lot of the time. I’d probably just fold on the flop in Villain’s shoes. But they continued, so maybe you just exploited them better than I would have 
That aside, I agree that a lot of players don’t have much Kx here in their turn check-back range (think about how much value you could capture in Villain’s shoes if you did! This is why one of the most common leaks is to always bet when you have a strong hand and only check with the weaker hands in range).
Overall I think you did read Villain like a book and extracted max value with QQ when all is said and done, but I’d maybe think about how different lines could lead to you winning more chips against the other parts of Villain’s range.
For instance, if Villain had AdQd instead, you’re probably just getting a fold on the flop where a smaller bet might have tempted them to peel with very low equity. And a larger preflop 3bet might have gotten AdQd to just 4bet shove pre, again in really bad shape
Don’t think about a specific card that your opponent may have but all the possible combinations he can have,
to add to @Punlsher’s very good note — you are the preflop 3bettor! On a K hi flop, Villain is scared of YOU.
This is another reason B100 is likely not represented with much frequency by solver here; it should be kinda difficult for Villain to continue with hands that QQ dominates when Hero uses this size on the flop.
I agree that a min bet preflop is generally not a good move and when I started here I was using 3x opens, but I found that 2x has been very effective here because of the type of players that I can only attribute to being low stakes play money tournaments. The ring games here do seem to require bigger preflop bets. Playing for real money I would definitely size up.
I also agree that the flop bet conventionally should be more like half-pot here but again I have found that in a low stakes play money tourney, folks love to call post-flop. Obviously if he actually has a K he’s not folding, but if he has Ax the big bet could push him off of his equity (an A on the turn or the river is just as bad for me as that K).
Also, bear in mind that I’m out of position to him.
I thought that I was beat. That’s why I checked the turn, fully expecting him to push me off, but then he checked back. No one would ever check that back with a K.
So I agree with you on bet sizing in general, but in these low stakes tourneys I have found that standard bet sizing is a little out of whack and I have to adjust them to the style of play.
Totally agree with you about the population tendencies.
But we don’t want to push Ax off their equity; those holdings are in really bad shape on the flop and I think we will profit more from using a size they can (perhaps incorrectly) continue against.
If we consistently use sizes that should unequivocally deny equity to marginal hands, we’re effectively forcing our opponents to play correctly.
Of course, and before that flop I’m thinking of his range, but after seeing that flop the only thing that I’m concerned with are K and A and maybe spades (which I do not block). Yes I suppose an 8 would be bad but there are only 2 left in the deck for him (and one of them actually rivered). I don’t want him to turn an A or a spade so when he calls, it’s really all about a K.
But not necessarily on this flop. The original raiser can have A8s, 78s and 89s (K8s can open but should fold, and I think that’s fairly likely). There’s isn’t much 8x in a 3-bet range, and KK should have shoved pre, so an aggressive opponent might just raise all-in here.
I don’t think there’s much reason to bet QQ on this flop. No better hand is ever folding, everything else is drawing close to dead. Also, if they check behind, you can be way more certain they don’t have a K (vs the check on the turn where they could be afraid of the flush or a better K). They’re not going to put you on a K either, so you will get called down on the turn and river by most pocket pairs.
While that’s true Guru in cash games, and in tournament’s as well, but less so then cash games, it’s important not to bet WAY too much and WAY to little in tournaments, as it’s important to extract value yes in tournament’s, but it’s also just as important to NOT get too greedy, give free, cheap cards to beat you, that could cause you to lose a lot of chips, or get knocked out of tournament, or create a harder decision later. If you win a smaller pot more safely, that’s often, but not always better then losing a big pot, in tournament play. Remember in cash games you can just put more money on table, and keep playing. But in tournament’s, once you bust out there is no recouping the lost money, chips, etc, like a cash game, unless its a rebuy tournament. This is why slowplaying, or betting extremely small on a extremely wet, drawy board is usually so bad. That same logic applies to Napkin’s hand. As Napkin said, Napkin was out of position, and does not want to see a Ace on turn or river etc. So Napkin wants to make hands pay too much to continue, but not bet too much to automatically force a fold. A better bet between what Napkin betted, and the small bet your advocating, would be between about 3/5 pot to 2/3 pot to 3/4 pot. That bet size can still semi get AQ suited to maybe call, peel, and pay way too much IF do so, as a big mistake, and can, semi probably will semi force a fold, which creates a win win scenario, because if villain folds, great Hero takes down a smaller pot, safely, and doesn’t have to face a A on turn, river. And if Villain makes a mistake, pays way too much, then Hero makes about a 3/5 to 4/5 pot bet on turn,if no A on turn, using the same logic, bet sizing, etc, as flop, as hero doesn’t want to see a ace on river, and if a ace is going to hit river, Hero want to make Villain pay WAY TOO MUCH. So again its a win win, as either Villain pays way too much, or Hero takes down a safer, smaller pot, and is still in tourny with a bigger stack. Remember tournament play is different then cash game play, so in cash games, there is more slowplaying, small betting with semi good somethings in cash games, then in tournaments, etc.
K88 fd is not a wet, dynamic board though. Trying to protect against the ace when you’re already losing to a K and double blocking AQ does not make much sense. They likelihood of the opponent calling with a big bet with naked ace, and hitting on the turn or river is pretty low.
Alternatively, you’ve folded out hands your doing well against, or made the pot bigger when they do hit an ace, so it’s lose/lose. I don’t think that’s a particularly useful way to think about the game, at least not in most spots.
If you’re betting for value and then are almost certain you’re behind when called, you’ve bet too big. I think that would be the case here whether you bet pot or 2/3rds. I think we all agree villain is paying WAY TOO MUCH if they call with worse here, so betting big and still being ahead when called is relying on our opponents to make a pretty big mistake.
You don’t think that K88 doesn’t have the potential to turn 2 pair or a set into a boat? Sounds dynamic to me.
There are only 2 cards in the deck that turn a worse pocket pair into a boat, and that doesn’t even change the nuts. It’s incredibly rare for anyone to be drawing dead on the flop, so there’s almost always some risk of getting outdrawn, but this is about as dry as it gets in a 3-bet pot.
Just an aside to all this great discussion - having played more hands in MTTs than anyone on the site this year (to my significant shame lol) I think it’s super important to keep a clear distinction between what works on Replay, and what we actually “should” be doing in theory.
You can definitely build a big stack all the way up to 250k buy in events on Replay just by having a Top 15% hand and more or less indiscriminately shoving large amounts into the middle. @AbuseMod is really good at executing this strategy and tbh it always makes me jealous when I’m over here grinding and range betting 1/2 pot and he’s just like “meh I have AJ I’m gonna shove 60bb over 2 EP limps” and he gets called by KJs and A9o and triples up lol
Still, even in the case of these huge deviations from equilibrium that prove to somehow still be profitable exploits because the field is so bad at poker, we should keep in mind what the mechanism is that we’re profiting from. Mostly, IMO, it’s not equity denial but absurdly thin value. We can profitably overplay hands as long as our opponents overplay their hands even more egregiously, which is awesome 
But we should be clear-eyed about what we are doing. Full pot with QQ on K88 two-tone is a massive overplay; I didn’t think this far ahead but I actually agree with @lihiue that it’s not even a bet at all most of the time at equilibrium. Villain overplayed TT even worse, though, so suddenly a mistake becomes a big win. If he’ll do this with TT why not with 99, 88, surely there’s a cutoff somewhere but you get the idea.
I just think it’s worthwhile to keep track of the difference between what is optimal at equilibrium and what works against the mostly “I would be a mild to significant loser at $1/3” Replay pool.
You have a point, but I disagree that a K a pair and a flush draw makes a flop dry. Lotsa stuff can hit that different ways.
Also, yes, absolutely, optimal play isn’t optimal at this level of competition. I think that we all agree that playing for real money or even playing play money in a ring, we stick to more conventional bet sizing.
However, what we also all seem to see is pocket 10s played this very poorly, and he got caught. 
If you keep going full pot OOP with QQ on K88 two tone, I think it’s gonna catch up to you, I guess that’s the tldr of it
There are better lines to take
We can justify a lot of plays if our opponents make howlers but that doesn’t make them the best decisions
Anyway, for me the biggest takeaway from this hand is how ridiculously profitable it is to check a strong but non nutted hand after previously showing aggression. Weaker players read way too much into discontinued aggression