It means that the curve $1/(k-k_0)$ has equal and opposite areas on both sides of the singular point $k_0$. If we break the integral into one over positive $k$ and one over negative $k$, a change of variable $k\rightarrow -k$ allows us to rewrite the last equation as $$\begin{equation*} \int_{0}^{\infty} \frac{dk}{k^2-k_0^2} =0. \end{equation*}$$