I think this is the key. UConn obviously recruits talented players, but they are also very selective in their recruiting, with (according to the accounts I have read) a lot of attention also being given to things like effort, body language, support of teammates, reactions when having a bad game, and so on. And then Geno + company get to work on the the "much more than that" in practices that are brutal and often set up to put players in impossible situations in which they lose more often than not.
I also think that Auriemma is extraordinarily skilled at reading people and that in turn makes him a master motivator, who is willing to do whatever he thinks it takes in order to make his players better, including encouraging them to "hate" him at times if that will make them more successful on the court. Usually, the greater a player's potential, the more he's on her case. (Stewart is the poster child for this, with Samuelson the heir apparent this season; I would imagine that, if DD had chosen UConn, their clashes would have been epic.) When it works, and it seems to most of the time, the results can be stunning: it's difficult for me to believe that Samuelson and Collier, as sophomores, are the same players who were hardly central last year as freshmen. When it doesn't, players transfer (about 1/year on average).