The literal sense of the Latin insulto, from which the word “insult” is derived, was to leap at or spring upon a person or thing.
A wild beast might spring upon its prey, ready to tear it apart; a soldier leap at his foe, prepared to take his life.
But the ancients realized that they could tear a person apart, in a figurative sense, by a torrent of abusive words, or kill him with scorn and abuse, and thus insulto acquired such a figurative meaning, giving rise to the sense in which we use insult.