A city's "art scene" is all of the art shows, galleries, events, artists, art buyers, and so on that happen there. A "scene" includes both the events and the people. It centers around a particular city or neighborhood, and also a certain period of time.
In addition to "the art scene", here are some other kinds of "scenes":
the 1980's San Francisco punk rock scene
the club scene in Miami
the New York fashion scene
People who belong to a scene tend to talk, dress, and act alike. Some people fit in really well within the scene. These people can be called "scenesters", although that's a little bit of a negative term.