A showing that proves rather than describes. A demonstration ends an argument by being watched.
Latin demonstratio (a pointing out, proof), from demonstrare (to show fully), from de- (thoroughly) + monstrare (to show), from monstrum (a sign, portent); a showing that proves rather than describes, ending an argument by being witnessed directly.
It shares demo's root in monstrum, an omen strong enough to need no argument, at full length rather than clipped. What a demonstration proves, it proves by being witnessed directly, the same way an omen was never argued for, only seen.