Most people have covered it, but to elaborate a bit...

In the studio musicians are rarely playing live with the rest of the band, and even if they are playing with the rest of the band they will often be isolated in their own room. The headphones are there so they can hear the rest of the music and/or click track/whatever else they need to hear. This is done because when recording music engineers generally want to get each instrument recorded separately so they can easily manipulate whatever they want on that instrument without causing any issues for the other ones. If, for example, we had a bass and drums in the same room, if one of them made even the tiniest mistake you'd have to re-record both instruments (the bass would be heard in the drum mics, and the drums would be heard in the bass mic(s)). If the instruments are separate then you could just re-record whatever little part you needed to. This also applies to any other treatment the engineer might do, compression, EQ, pitch/timing adjustments..etc. Its basically always better to have the instruments separate for most modern recordings. Things like orchestras often record as a group because the interaction of that many instruments in a room sounds different than just playing each instrument separately and playing them back at the same time, and most classical recordings aren't going for the same level of inhuman perfection that most pop/rock/other modern genres are.

For live performances it's because of a few things, but mostly 1) stage monitors kinda suck, and 2) stages are often really loud. In ear monitors allow musicians to hear everything they need more clearly and at a significantly lower volume than without.