According to Tadashi Tokieda, math is creating mental images. If you are not very mature, you can probably still make sense of Geometry, because it consists of images. Then you go more and more abstract and have to paint mental images of things that aren't actually visualizable.
As to anecdotes: I had a moment of feeling mathematically mature when I read about the fact that SL(2, C) is not compact and thought: Yeah, how could it be, it can transform squares into parallelograms of arbitrary diameter and can therefore not be bounded. This is one case where I just had this mental image which lead to immediate understanding, no proof required. Felt like a real champ that moment.