I like to watch the openings of anime and other TV shows on YouTube. I find it stupid how companies would have the audio silenced or the video removed completely. I sometimes choose what shows I want to watch based on what their TV openings look like, because if an opening looks boring, why should the show be any different.
The same thing applies to animated music videos. I see AMVs as a gauge of a show's health. If nobody is making AMVs for a show, then the fandom is not doing well. If there's a lot of AMVs, then there are a lot of people interested in the show enough to make a little tribute to that show. It also has the benefit of promoting an musician's work. I've gotten a few songs from artists I would otherwise not have listened to if it hadn't been for AMVs.