What caused the two families to become bitter enemies?
This is a great question! We do not know. Stories all have to start somewhere, and every story has things that have likely happened before them that impact what happens during them.
The prologue states
From ancient grude break to new mutiny.
The word ancient makes us know that this has been going on for a long time. Sometimes feuds like this begin and those fighting no longer remember what caused their competition.
The prologue also tells us that the course this story takes will result in them not fighting any longer.
We have no idea why the two families in Romeo and Juliet come to be such bitter enemies that they are having this feud. All we know about it, really, comes from the Prologue to the play. There, were are told that the hatred has been going on a long time. We are also told what the result of the hatred will be.
However, we are not told how it is that the enmity started. So we end up with no idea as to why they hate each other.
Luckily, it does not really matter -- the cause of their feud has no real bearing on the story itself.