Morera's statement says that if $f: D \to \mathbb{C}$ is continuous on some connected open set $D \subset \mathbb{C}$, and $\int_{\gamma} \ f = 0$ for any closed piecewise $C^1$ curve $\gamma \subset D$, then $f$ is holomorphic on $D$.
In this case, $D = \mathbb{C}$, and it's clear that $\int_{\gamma} \ f = 0$ for any $\gamma$ which does not intersect the unit circle. If $\gamma$ does intersect the unit circle, then we can argue by continuity of the integral that $\int_{\gamma} \ f = 0$ still (split up $\gamma$ into finitely many parts, each of which intersects the circle, and deform each 'infinitesimally' so that each part no longer intersects the unit circle).
Now we are done.