Scientists occasionally discuss how a zombie virus could possibly work. According to this article, the most likely explanation is that a zombie infection would not be a virus at all, but a prion disease. Prion diseases involve malformed proteins that share the same chemical structure with their biologically active counterparts. These diseases are caused by the malformed proteins corrupting the structures of the biologically active ones.
The first prion disease discovered was kuru, which is spread by eating brains. Because, of course, eating brains has to have something to do with it!