As others have pointed out, we do it because it lets us hear things better. What I haven't seen pointed out (could have missed it) is that as musicians we generally don't hear what you think we hear. Either in the studio or on stage, each of us wants a unique mix. For example, I play keys (a percussive instrument) so I generally want more drums and bass, with less rhythm guitar and even less lead guitar. I want to be able to hear them for cues, but they're super quiet in my mix. The only way to do this for every musician is for every musician to ONLY be able to hear their own mix, so you use headphones (studio) or IEM's (stage). If the audience heard what I heard, they'd probably think the band sounded awful, but that's what I need to be tight - especially during improvisation or complex segments that require syncopation / doubling / whatever other fancy stuff my guitar player is addicted to that day.

The alternative, if you're wondering, is that everyone's mix on stage bleeds into the other mixes - which leads to a lot of noise competition as the night goes on and hearing goes out.