is holomorphic everywhere except its poles. And the contour doesn't have any singularities / poles on it.
Now, think of the Laurent Series of f(z) on the point a.
Integrate both sides by the contour ,
Cauchy's integral theorem tell us that for any n except -1.
So it means all integrals except the red one will vanish (because they are 0).
Now we have,
Let be a circle of radius 1 with center in the complex plane.
This will give,
The -1st coefficient of Laurent Series of f(z) at a point is called the Residue of that function at that point.
So we can generalize this for contours with multiple poles,