When someone takes umbrage at something, they find it offensive, and it probably makes them angry.
Umbrage comes from the same source as umbrella, the Latin umbra, "shade, shadow." The umbrella was invented to keep you in shade, and when you take umbrage at something, you're casting a shadow over the person or thing responsible for the offense. I take umbrage at the suggestion that I'm not a nice person: it's offensive and infuriating. After having devoted my life to helping animals, I might take umbrage at the notion that I've been doing it for publicity purposes: I resent the idea that it was for any other reason than my love of animals.