In the particular case of surfaces, I found the following reference which includes a proof that is not too complicated: *Regular Mappings and the Space of Homeomorphisms on 2-Manifolds* by Hamstrom and Dyer. It works for surfaces with or without boundaries and includes a slight generalization with fixed points on the boundary. This is Theorem 1 in this article. The proof fits in 6 pages, is a bit heavy on notations but this remains managable. Unfortunately there is no figure. The proof uses conformal maps for a couple of lemmas, Alexander's trick, and a technique due to J.H. Roberts but I cannot find the corresponding reference. They also cite a German article of Kneser as a source of inspiration, but my knowledge of German is very basic so reading it would represent quite an investment.