Your question refers to the opening speech made by the Chorus, which tells us about the state of affairs in Verona and in particular draws our attention to the feud between the Montagues and the Capulets, and how their "ancient grudge" has broken out in a "new mutiny." We can see that the subsequent tragedy of Romeo and Juliet is actually, at least in part, a result of this "ancient grudge" by analysing a key scene of this play.
In Act III scene 1, having just married Juliet, Romeo runs into Tybalt. The "ancient grudge" that has created a seemingly permanent emnity between the two houses means that Tybalt tries to pick a fight with Romeo. Romeo, being aware that now Tybalt is actually his kinsman, refuses, and so Mercutio fights with Tybalt instead, and is killed. Romeo is now honour bound to gain revenge for the death of his friend, Mercutio, and does so, killing Tybalt. We can see in this chain of events that the "ancient grudge" had a vital role in the subsequent tragedy that befell the two young lovers. Were it not for the feud between the two households, Romeo would not have killed Tybalt.