Why do people make them? For fun! A lot of the AMVs are intended to be humourous, while some are meant to profile a character etc. It can be hard work, but very satisfying. As an example, this is Hsien Lee's AMV that puts Gunsmith Cats to the theme from Magnum P.I. (Right-click & "save target as"):

Gunsmith P.I. AMV (14.7 Mb, Can be played by Windows Media Player)

  This AMV is one of my personal favourites, made by David Appelman - Initial D to "Greased Lightning" from the Grease soundtrack (Right-click & "save target as"):

Initial D Greased Lightning AMV (18.7 Mb, Can be played by Windows Media Player)

  If you want to find AMVs, or would like more information about making them, the best place to go is http://www.animemusicvideos.org.