My understanding is that it comes from beat poets/ranters that speak into a microphone, wrap up their bit with a killer line or question, then just extend their arm and “drop the mic.” By that moment, the audience is either stunned into silence or cheering wildly. The microphone makes an audible “boom” as it hits the floor. The speaker leaves the stage immediately after the drop with a quick turn to the wings or backstage and walks away, ignoring the audience.
Well, “dropping the mic” is just another meaningless expression that’s kind of stupid, and those of us with a modicum of intelligence should try to stay away from stupid and promote clarity.