Origins
When you insult someone, you are — in the word's oldest sense — leaping on them. Latin insultāre meant 'to jump upon, to trample', from in- ('upon') and salīre ('to leap'). Roman victory celebrations included physically leaping on the bodies of fallen enemies.
The Latin root salīre produced one of the most surprising word families in English. Assault is 'leaping at'. Result is 'leaping back'. Exult is 'leaping out' with joy. Resilient is 'leaping back again' — bouncing back. Salient means 'leaping forward', the point that jumps out at you.
Latin Roots
Most unexpectedly, salmon belongs to this family. The Latin salmo meant 'the leaper' — named for the fish's spectacular jumps upstream.
The shift from physical to verbal happened through medieval French. By the time insult entered English in the 1540s, the leaping was entirely metaphorical. But medicine preserved the physical sense longer: an insult to the body — a wound or trauma — remained standard medical terminology into the 20th century, and still appears in clinical writing today.