The history of mime should be pretty easy to look up. I expect it’s quite an ancient art.
Perhaps (just guessing here) it originated as entertainment form in settings where there was no sound amplification. Or (just guessing again) it has its history in religious rituals where the words were best left to the gods’ authorized mouthpieces. Or both. In Hamlet, the play-within-a play actors mime the play before acting it out with dialogue.
Mime may use some symbolic forms and gestures, as do, say, ballet, hula, and puppet shows, but I doubt that mime itself symbolizes anything.