stslavik's answer covers a lot of the sciency benefits as they relate to learning and such.
I'd like to add, from the sport psychology / coaches point of view:
Like a lot of other things, practicing in as close to competition environment / circumstances as possible will provide a greater ability to perform at the same level in competition as you do in practice. So, if there is music playing at your competition venue, it makes sense to have it at practice.
Music also helps set the tempo of the class. I find in my bjj classes we often listen to reggae style music, it keeps people chill and seems to keep people from being a spazz and going to hard. In Boxing and MMA, we tend to listen to heavier rock and roll, alternative and metal. I find if i'm hitting the bag, listening to disturbed, or metallica, i hit the bag harder and my strikes are crisper then when i'm listening to bob marley.