Help in the proof of the Laurent series theorem

Question

I'm reading Conway's complex analysis book and on page 107 he proved the following theorem:

I didn't understand this part of the proof:

Why $f(z)=\frac{1}{2\pi i}\int_{\gamma} \frac{f(w)}{w-z}dw$?

What the winding numbers $n(\gamma_2,z)$ and $n(\gamma_1,z)$ have to do with everything?

Answer 1

On page 92 he stated the following theorem:

Now it suffices to use this corollary to have the hypothesis of this theorem on page 84: