When Andy nearly gets shot-gunned attempting to marry the son and daughter of two feuding families, he decides the best way to help the young lovers is to find out what caused the feud in the first place - and, if possible, resolve it. After explaining to Opie and Aunt Bee the dangers inherent in getting caught between feuding families (via a home spun version of Romeo and Juliet), Andy sets out to end the long running feud.
However, despite his best efforts, Andy is unable to discover the cause of the feud - because no one seems to remember it! All he learns is that they're feudin' "because they're shootin' at one another" and that "they're shootin' at one another because they're feudin'." Furthermore, he discovers that despite decades of so-called "feudin'", no one from either family has ever been injured! Andy attempts to shame the feudin' parties into peace by pointing out how bad it makes them look to have a feud where no one ever gets killed. He then prods the two fathers into a half-hearted, fake duel - which sends both of them running for their lives!
When it becomes clear that the two youngsters have much more courage than their elders - by being willing to stand up for each other - both fathers realize the potential of their offspring and embrace the marriage, ending the feud once and for all.