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This theorem was proved by Hurwitz in the 19th century, who in fact showed the stronger theorem (also mentioned in Danny Ruberman's answer) that any finite-order diffeomorphism of a surface of genus at least $2$ acts nontrivially on homology. This is proved in the same paper where Hurwitz proved the more famous Riemann-Hurwitz formula. I give an exposition of what is essentially Hurwitz's original proof in my note "The action on homology of finite groups of automorphisms of surfaces and graphs", which is available on my webpage here. This proof is not as efficient as the proof using the Lefshetz fixed-point theorem that Danny mentions, but it is very direct and elementary (basically only using the definition of homology).