I don't think this thing called "intuition" is all that mysterious.  It seems to me it's just a result of extensive training of the neural network we call carry underneath our hats.  Study a lot of chess games, and you'll absorb a lot of chess knowledge, even if you can't output that knowledge in the form of a set of rules.  Really, it reminds me a lot of how after a few years of foreign language study, a lot of the grammar rules one would learn in a first-year course fade away into the background and become more of a gestalt knowledge of basic language forms (which is really just a fancy-shmancy way of saying "you know what sounds right and what sounds wrong.")