Magneto’s whole shtick is that he survived the holocaust. This made him a radical against discrimination. So when he sees the mutant minority starting to get the same treatment as Jews in Germany (dehumanizing propaganda, lists and “yellow stars”) he’s like “not this time” and decides to start fighting before things get concentration camp bad.

Professor X believes that regular humans and mutants can reach an understanding despite all the propaganda to the contrary. But if mutants on one side (magneto)attack then due to the powers only other mutants are really effective, both for stopping them and for showing that it’s “not all mutants”.