It is rare for artists, magicians, and other professionals to let people see how they work. Mathematicians, too, rarely expose the thought processes that led them to a discovery – not necessarily because they want to keep this a closely guarded secret, but often just because they just don’t know what the process was (or find it very difficult to describe). Poincaré famously wrote about the mysterious mechanisms of mathematical discovery, emphasizing how letting the subconscious take over helps magical insights to reveal themselves. This is also manifest in how people tend to present proofs: in almost all cases, the argument is presented in the most elegant and slick way, which usually implies that the motivations, partial ideas, false turns and mistakes are well hidden from view.
Tim Gowers is one of very, very few world-class mathematicians who are both capable and willing to share the way they think about problems (the other one I know is Terry Tao). In a recent series of posts (starting here), Gowers explains his attempts to attack the problem of finding lower bounds for circuit complexity. They are fascinating both as a great way to understand the various negative results that were proved in this direction (Razborov-Rudich-style, on why certain natural approaches are doomed to fail), and as an opportunity to see how Gowers approaches the problem.
I agree – its amazing the kind of transformational thought that Timothy Gowers, Terence Tao and the like seems to bring to this world. Yes, brainwaves and unusual insight and hard work and seclusion
forms the core of mathematicians and is considered unusual and looked upon with awe by laymen like me and others BUT its a whole lot more valuable when they contribute selflessly to a bigger and grander purpose of acknowledging that SUM of Parts is bigger 
and reach out to source of knowledge unbiasedly. They make their prowess so humbling and achieve greatness.