While I can blindly follow the equations and come up with solutions that are right I still can't wrap my head around WHY if you multiply the resistance and the capacitance you arrive at a number (tau) that just happens to equal the time it takes to reach 63% of the maximum voltage.

It just doesn't seem intuitive and it's bothering me. Is it simply too complicated for a beginner to understand so it's not explained in depth or am I missing something?

http://www.kpsec.freeuk.com/capacit.htm#charging