A violin/viola mute is a small rubber piece that can be attached to the bridge of a violin/viola to dampen the sound of the strings. The rubber restricts the vibration of the strings giving it a darker, lower sound. I don’t really think people use it for much else, it is rubber though so I’m sure it could be of good use in an impractical way. I used to use the mute to practice my viola at home and also whenever it was needed in music when I played in orchestra and pit band.
In Phantom of the Opera’s, Angel of Music backstage scene in Act 1 the strings use mutes after the tremolo while Christine is singing. It says mutes on the 3rd beat of the first measure. In measure 11a it says “con sord”, which means “with mute” in Italian, this tells the musicians to put on their mutes. I believe the purpose of the mute is to make a darker and somber sound especially with this being a moment with Christine and the Phantom it’s meant to be a quiet and connecting scene. This adds mystery and emotion to the scene.