What insight is supposed to be gained from this complex analysis exercise? Let $C_0$ denote the circle centered around some point $z_0\in\mathbb{C}$ with radius $R$. We can parametrize this circle like this:
$$\begin{array}{cc} z(\theta)=z_0+Re^{i\theta}, & \theta \in [-\pi,\pi]\end{array}$$
We are asked to show that:
$$\int_{C_0}(z-z_0)^{n-1}dz = \left\lbrace \begin{array}{ll} 0 & n\in\mathbb{Z}\backslash\{0\} \\ 2\pi i & n=0\end{array}\right. $$
I have completed the exercise but do not quite understand what this gives us. Could someone please provide some form of intuition for this?

Consider the function $f(z)=\sum_{-\infty}^\infty c_n z^n$. Your result tells us that if we want to determine the coefficient $c_{-1}$ we just need to integrate $f(z)$ over a closed contour about the origin.
$$  \int_C f(z) dz = 2\pi i c_{-1}$$
If we want some other coefficient we can multiply by some power of $z$ first and then integrate. For instance integrating $z^2f(z)$ will allow us to determine the coefficient $c_{-3}$. 
All of this hinges upon the fact that the integral is nonzero only when $z$ is raised to the $-1$ power. Another way of writing this is,
$$ \int_C z^n z^m dz = \delta_{n+m,-1}2 \pi i \equiv \begin{cases} 2\pi i \text{ if } m+n=-1 \\ 0 \text{ otherwise }\end{cases} $$ 
this is what is usually called an orthogonality relation and extends to applications outside of complex variable theory. This property provides us with a method for determining the coefficients for the expansion of our function in terms of some family of orthogonal functions. Some examples would be the fourier series expansion, hermite polynomial expansion, and the spherical harmonic expansion.
