Why is it important to the story that the prince broke up the fight ?

The Prince breaking up the fight is a dramatic way for Shakespeare to tell us the recent backstory or history of the feud between the Capulets and the Montagues. When the Prince breaks up the fight, he is angry. He scolds the fighters. He tell them that the two families have three times now engaged in street brawls that have stirred up old quarrels. These brawls are disturbing the peace. They are dredging up old hurts. The Prince puts it, using colorful language, as follows:

Three civil brawls, bred of an airy word,
By thee, old Capulet, and Montague,
Have thrice disturbed the quiet of our streets
And made Verona’s ancient citizens
Cast by their grave-beseeming ornaments,
To wield old partisans in hands as old,
Cankered with peace, to part your cankered hate.

The Prince has had it. He is sick of all this fighting. He tells the Montagues and the Capulets that if they don't cut it out now and stop the nonsense, he'll have anyone who starts a fight put to death. He says:

If ever you disturb our streets again,
Your lives shall pay the forfeit of the peace.

This speech explains the problems Romeo and Juliet will face as members of feuding families. We also understand why Romeo's life is on the line when he kills Tybalt for killing Mercutio. Because of the Prince breaking up the brawl and making this speech, we understand how upset he is—and how determined to stop the violence. Therefore, when Romeo flees Verona, we understand why.