When a holomorphic function between hyperbolic surfaces is a covering map.

I'll follow the proof as it is in Milnor's book. So we have Poincaré metric on the surfaces and the covering maps are Riemannian covering.

If $F$ is a conformal isomorphism, then $f$ is a local isometry.

Now, just use that a surjective local isometry with complete domain is always a Riemannian covering map.

Observation: This fact about Riemannian geometry used in the end can be found on Manfredo's book, for example, on the section about Hadamard theorem. The precise statement is the following.

Let $M$ be a complete Riemannian manifold and let $f:M \to N$ be a local diffeomorphism onto a Riemannian manifold $N$ which has the following property: For all $p\in M$ and all $v\in T_p M$, we have $|df_p(f)|\geq |v|$. Then $f$ is a covering map.