partir is intransitive in modern French ("depart") but primarily transitive in Old French ("distribute", i.e., "make depart"). The transitive meaning is still preserved, as an archaism, in the set phrase "avoir maille à partir avec...".
Analyze the prefix separately from the root:
re- - again, anew.
partir - distribute (blows).
During the 17th century, fencing was not yet a sport, it was a deadly combat activity. The activity consisted of distributing blows to opponent(s) without receiving too many.
The prefix re- in "repartir" is actually etymologically the same prefix as that of English "reply" (from Latin "replicare" through Old French) and riposte (from Latin "respondere" through Italian and French), so that morpheme's various reuses (pun intended) are hopefully intuitive enough even today.
The word "répartie" with the sense of "reply" existed already in French. It already lost some of the "thrust for a thrust" connotation in current French, so it does not have to be a particularly witty reply or to be replying to an immediately preceding verbal attack in French, unlike "repartee" in English. So while both the English "repartee" and French "répartie" can be translated as a "reply", the former's meaning is more specialized.
Note that "riposte" itself has at least two quite distinct meanings in English, I'm not quite sure which one you were referring to.
- In fencing, it's an counter-attack immediately following a parry. It has special implications within the rules of the sport.
- In a verbal exchange, it's a reply to an insult or to another verbal attack.
The second meaning is metaphorically derived from the first one. Etymology online for riposte: "Sense of "sharp retort; quick, sharp reply," is first attested 1865. As a verb, 1851."
TL;DR: An earlier meaning of replying to physical blows, i.e., answering distribution of blows with "distributing back", was metaphorically extended to an exchange of verbal blows between trained brains.