The beathook is quite simply a tool for DJs. It is no more than a piece of molded plastic in a perpendicular shape, with a curvature on the shorter part that comes out the middle. You slide the long part in the pocket of your jeans and hang your headphones next to your butt when you're not wearing them. Before this device was invented, DJs would either keep their headphones on their neck (which would get hot and uncomfortable in a crowded club) or lay them on their table, which is usually full of records and other crap.
It costs about eight bucks and comes in four different colors to choose from.