When I speak about intuition behind the proof of the functional equation. I am talking about proofs similiar to this one http://www.math.harvard.edu/~elkies/M259.02/zeta1.pdf
As far as I can tell, this idea was originally formulated out of necessity. Reimann Riemann needed it for the prime number theorem. However the intuition becomes more natural ifaccepts if one accepts both these of the following facts:
1)The Cahen Mellin Integral transforms a dirichlet Dirichlet series (something hard to work with) into a fourier Fourier series/polynomial (which we usually are more familiar with).
So in a sense it is the classic scheme of "I can't work with A so I will transform it to B and then transform it back." When you transform the $\zeta(2s)$ you are getting a fourier Fourier series (which by itself has a ton of facts about it) but it also has a really useful functional equation to it.