Some musicians (particularly drummers and those performing near to them) wear ear plugs to prevent hearing damage.
But some musicians also have earpieces which let them hear the performance as the audience (or recording equipment) is doing, rather than only hearing what's going on around them. So, for example, a drummer can tell whether he's playing too hard or too soft, or a singer can adjust their tone and inflection to meet the rhythm of the music.
I'm no musician but I've done some presentation work for local radio. People often ask why people in radio wear headphones, even when they're only interviewing a person directly in front of them, but I can tell you that it becomes automatic to change your voice slightly in response to the output which you're hearing through your 'cans'.
Chris