When I started studying mathematics, I soon started to wonder what kind of superhumans make the mathematics up, for the mathematics was full of bizarre definitions, and unreadably long proofs. There was also some devilishly clever short proof, but they did not make it easier, for I knew that one had to posses divine intelligence to come up with them. For example, I did not understand how a mortal could read, much less invent, a statement of a theorem spanning three pages or its proof.
However, behind every definition, every theorem, every proof, there is an idea. Often the simplest implementation of the idea does not work, and the original approach has to be modified or supplemented with technical conditions. When the correct piece of mathematics is found, the original idea is buried deeply. Hence, explaining the idea involves far more than writing down logically correct mathematical statements. A good explanation requires description of the whole process of the discovery.
The process of discovery is always messy, and it is tempting to reveal as little of it as possible. An honest description of the convoluted twists of mind that culminate in a discovery is both long and embarrassing. The discoverer is thus tempted to impress the reader with the shortest argument possible. Since it is rarely possible, monstrosities, dense with equations, are born.
Natural mathematical explanations that are also well-written are thus extremely rare. Yet they are indispensable for appreciating, and learning mathematics. Below I have listed links to natural mathematical explanations that I know of. The topics they treat are wildly different, but the authors of all of them have managed to convey the joyful process of discovering mathematics. Please, suggest more...
The ultimate goal of mathematics is to eliminate any need for intelligent thought. [Alfred N. Whitehead]