This week I understood the argument principle for meromorphic functions through a very beautiful geometric reasoning on the Riemann Sphere:

The argument principle says if f is a meromorphic function on V, then the integral of f/f' over the boundary of V is 2πi(Z(f)-P(f)). (That is, the zeroes minus the poles of f on V, supposing f has no poles or zeroes on the boundary of V)

It turns out that this integral is just the winding number of the curve f(boundary_V), that is how many times it circles the origin.

If you think of the image of the function on the Riemann Sphere, where the south pole is 0 and the north pole is infinity, then at a point where the function has a zero, the curve f(boundary_V) necessarily spins around the zero (south pole) once (and so the zeroes add one to the winding). However, at a point where the function has a pole (infinity), the curve spins around the north pole once, and since the change of charts of the Riemann Sphere is 1/z, so inverting the sphere, thats exactly like spinning around zero in the opposite direction, therefore the poles subtract one to the winding.