I assume you mean simply connected rather than "simple", and also that you mean $\mathbb C\setminus \Omega$ rather than the other way around.
In that case it is because by definition of "simply connected" any closed path in a simply connected region is homotopic to a point. The winding number is a continuous function of the curve, and since for a closed curve it is always an integral multiple of $2\pi$ the winding number cannot change during a homotopy, provided that it exists at every intermediate step -- such as here when $\alpha\notin \Omega$.
Since a point (i.e., a constant curve) obviously has winding number 0, so has any closed curve that is homotopic to it.