I am trying to understand Cauchy's Integral Theorem which states
$$ \int_\gamma f(z)\,dz = 0. $$
If function $f(z)$ is holomorphic (has no singularities) within the area contained by the contour $\gamma$. I understand the proof comes from Green's theorem, but I don't understand conceptually why this is true. What exactly does the complex contour integral measure? It's not area, is it?