Why does the Gamma-function complete the Riemann Zeta function?

One way to get started is to look at the integral for the gamma function: $$\Gamma(s) = \int_0^\infty t^{s-1} e^{-t}\,dt$$ Subsitute $t=nx$ in the integral to arrive at $$\frac{\Gamma(s)}{n^s} = \int_0^\infty e^{-nx}x^{s-1}\,dx$$ which we then sum up to get $$\Gamma(s)\zeta(s)=\int_0^\infty \frac{x^{s-1}}{e^x-1}\,dx$$ which already shows that there is some connection between the gamma and zeta functions, and it does in fact allow us to extend the definition of the zeta function into the critical strip.

What comes next is far less obvious, but the idea is to introduce a branch cut for $x^{s-1}$ along the positive real axis, and to replace the above integral by one running from $+\infty$ along the bottom of the positive real axis, around the origin, and back to $+\infty$ along the top of the real axis. This introduces an extra factor $1-e^{2\pi i s}$. Now start expanding the circle around the origin, taking account of the poles of the integrand along the imaginary axis as we go, and end up with $$\Gamma(s)\zeta(s)=(2\pi)^{s-1}\Gamma(1-s)\sin(\tfrac12\pi s)\zeta(1-s).$$ From there, some cleanup still remains. As I said, this is not terribly intuitive, so it doesn't answer your question, but the first paragraph should at least give you a notion how the gamma and zeta functions are interrelated.

To the best of my understanding, the answer is yes, and this uniform way consists of some integration over the local field. This is explained in John Tate's dissertation. One starts with a certain smooth rapidly decreasing function, for which one takes the characteristic function of the p-adic integers in the nonarchimedean case and the function $e^{-|x|^2}$ for an archimedean field. This is being multiplied with $|x|^s$ (approximately) and integrated over the Haar measure of the additive group of the field. This produces the $\Gamma$-factor for an archimedean field and $(1-p^{-s})^{-1}$ for a p-adic field.

As has been explained above, the zeta function has a factor for each completion of $\mathbb{Q}$. The factor at $\mathbb{R}$ has to do with integrating $e^{- \pi x^2}$ and the factor at $\mathbb{Q}_p$ has to do with integrating the characteristic function of $\mathbb{Z}_p$.

Some people might wonder why these two functions were chosen. The answer is simple: they are both their own Fourier transforms.

Also, I don't think anyone has recommended Terry Tao's expository post on this material yet. It is quite good.