The reason for value betting at a basic level is one thing, to get worse hands to call. I look at the range of villians hands and compare that with the range of hands they are calling here (I go crazy with calling range notes). Then I compare that to my actual hand when deciding to bet.
Then it gets math time, how often does villian ahead of my value-bet range and calling, how often is he behind it and calling, how often are they raising as a bluff, how often are they raising for value (the 5th outcome of folding is irrelevant, win whatever is in the pot now in all cases). If we aren't calling a raise then him raising for value/bluff are basically one in the same. I weight each outcome with the chance of it happening and calc the EV, when I am 0 and greater than 0 I bet, less than 0 I check back. (bet size is a whole different thing and something I am bad at).
Lots of work off the table is required for this because players have very different calling ranges for the exact same situation. I basically use my past calcs to best guesstimate the EV of betting during real time. In this specific hand it gets trivial easy to value bet this river since we have the effective nuts, it gets much harder for very thin bets.
Sometimes you just lose more money with top two or something similar to bottom set because they call three streets but will be very profitable in the long run against his calling range which could even be bottom two from some very weak players when draws miss.
Let's say we didn't have the clubs and instead had A
A
7
6
. The hand plays out the exact same except that the river is K
, would you value bet that? Probably not, a large portion of villian's calling range is going to be flush draws and sets. Sets/2pair will probably be check folding that river to our bet once the flush gets there and the majority of villian's calling range is going to be what little 4-5 that didn't raise turn and flushes. Hence the bet is minus EV and I would just showdown.
Feel free to call me out if you think everything I've typed is BS but this is what I've done when learning to bet or to check behind basically for every street not necessarily just the river. Things do get different when you have to worry about bluff raises, then you have to add some value to checking behind but this seem perfect to me when people play straightforward.