The reason for the enmity between the households of the Montagues and the Capulets in Shakespeare's Romeo and Juliet isn't explained anywhere in the play. There aren't any tantalizing clues, no subtle hints, and no hidden meanings in the dialogue as to why or how the feud began.
Why the families were feuding seems to be of no consequence whatsoever to Shakespeare. The only thing that matters to him, and to the play, is that the families are feuding. The feud drives the play forward from the very first scene to the tragic end of the star-crossed lovers who get caught in the middle of it.
CHORUS: Two households, both alike in dignity,
In fair Verona, where we lay our scene,
From ancient grudge break to new mutiny . . . (1.Pro.1–3)
It's an "ancient grudge." That's the only thing we're told about the feud. How did it start? Nobody says. There's no discussion about it. Maybe nobody even remembers how the feud started.
Perhaps the feud started in much the same way that the recent brawling in the streets started:
PRINCE: Three civil brawls, bred of an airy word
By thee, old Capulet, and Montague,
Have thrice disturb'd the quiet of our streets . . . (1.1.85–87)
"Bred of an airy word" could mean a verbal slight of some kind from a member of one family to a member of the other family, or an insult—real or imagined—or something spoken purely in jest, but taken the wrong way.
Even at the end of the play, when the Montagues and Capulets reconcile—and when we might expect somebody to tell is how the feud began, now that it's ending—not one word is spoken about the reason for the feud:
CAPULET: O brother Montague, give me thy hand.
This is my daughter's jointure, for no more
Can I demand.
MONTAGUE: But I can give thee more;
For I will raise her statue in pure gold,
That whiles Verona by that name is known,
There shall no figure at such rate be set
As that of true and faithful Juliet.
CAPULET: As rich shall Romeo's by his lady's lie—
Poor sacrifices of our enmity!
PRINCE: A glooming peace this morning with it brings.
The sun for sorrow will not show his head.
Go hence, to have more talk of these sad things;
Some shall be pardon'd, and some punished;
For never was a story of more woe
Than this of Juliet and her Romeo. (5.3.307–322)
We'll never know how it all started, or why it came to this.