The Prologue merely refers to the feud between the Montagues and Capulets as an "ancient grudge" with no further explanation as to why the two families hate each other. We learn too that the feud has broken into "new mutiny."

As the play opens, we get further confirmation that the feud has recently renewed in intensity. After Prince Escalus breaks up a street brawl, he notes that fights have broken out three times recently between the families and disturbed the peace, or as he puts, it the "quiet of our streets. He is clearly fed up, because he decrees that if they ever disturb the streets again, whoever does so will face the death penalty ("your lives shall pay").

 One of Shakespeare's points is that the violence and feuding that ends with the death of Romeo and Juliet is senseless. Whatever might have started it, it no longer serves any purpose: the reasons simply don't matter. As the play demonstrates, hanging on to grudges is destructive. It is only after the shock brought on by the death of these two young people that the families are jolted out of the cycle of mindless violence that has controlled them.

Shakespeare never explains why the Capulets and Montagues hate each other and are constantly fighting.  

A reader can make a few educated guesses though.  The reader does know the feud has been going on for a very long time, and the reader knows that the feud is well known to the public. The reader also knows that both the Capulets and Montagues are wealthy and influential families in Verona.  Those facts lead me to suspect that the feud originated over some kind of economic dealings that went badly.  It makes sense that because both families are rich, they would each be upset over losing money or being out competed for more wealth.  It could be a political deal that went sour too. If both families are wealthy and influential, it makes sense that they are active in politics and city government as well.  Perhaps one family took advantage of a political ally in a way that hurt the other family. 

Or maybe it was a forbidden love between two star crossed lovers of long ago.