Argus Filch is the caretaker of Hogwarts. He also absolutely hates children. He is a Squib, like Mrs. Figg, but since he spends most of his life surrounded by gifted wizarding children, his hatred is probably the result of bitterness and envy. Filch has always made it clear that he would like to discipline the kids more firmly for their rule-breaking, but Professor Dumbledore never lets him.
To Filch, Professor Umbridge is like a dream come true. She has given him more authority to punish students using brutal, old-fashioned methods. In fact, when Fred an George set up their swamp, Filch eagerly presents Professor Umbridge with an Approval for Whipping to sign, so that he can actually beat the twins. Of course, he never gets to go through with his threat, since the twins fly away from Hogwarts to start their new business. But the fact that he wants to beat them so badly suggests that he will do anything to feel like he has power. As a Squib still working in the wizarding world, though, it's no wonder that Filch will seize any opportunity to feel in control of the magical people around him. Even if it means small-minded bullying and harassment.