why do musicians have an ear piece?

Monitor, it has the whole band, usually in a slightly different mix than the audience hears, so the musicians can hear the cues.

In the day the monitors were rather large speaker enclosures on the floor of the stage in front of the microphones. The earpieces, along with wireless cable instrument transmitters, allow for more freedom of movement on the stage.

I think it's also for the sound. It's very loud on stage where they are and if they didn't have that would prolly blow their hearing. They have to have some relief up there performing. And esp the singers that scream. And with the amps at high volume, etc it's extremely loud. It's loud for the crowd too, but louder on stage. It prolly echoes in their ears up there.

Your voice sounds different to you than to others because it resonates through your jaw bone. Also, the whole band sounds different than just one person. There are a variety of other reasons that others already listed as well.

sometimes a song-line is "fed" to them