Comes from the term "gig". It means you are moving from one gig to the next, singing along with each band for a few songs and then moving on. It's similar to clubbing, except the clubs are more likely to be coffee shops and there will probably be more singing and less dancing/drinking. Gigging is about experiencing the music and atmosphere of each venue. Tends to be more popular with people who enjoy jazz or "underground" music.
Me (PE major) - "Hey bro, want to shoot some hoops with Derek later?"
Friend (Jazz Guitar major) "Naw, I'm going gigging with some guys from my Theory 2 class."