The Prince is angry with these two men because of the feud that has been going on between their two families.
You could say that it should not be the Prince's business if the two houses are feuding, but it becomes his business because of the way it spills over into the streets. In the first scene of the play, we see the servants from the two families brawling in the streets. From what the Prince says, it appears that this has happened before. This disturbs the public peace and makes the Prince very angry.
The prince is angry with the Lords Capulet and Montague because they are the patriarchs of the two feuding families which,over a short period of time, contaminate Verona with brawls, death and tragedy. When he declares in Act 1 Scene 1:
What, ho! you men, you beasts
That quench the fire of your pernicious rage
With purple fountains issuing from your veins!
He is implying that the Capulets and the Montagues who are aristocratic and prominent families of Verona should set exemplary example instead of brawling like savages.
The prince is especially angry because the brawls not only,"Have disturbed the quiet of our streets", but has in Act 1 Scene 1 Lines 91-3:
...made Verona's ancient citizens
Cast by their grave beseeming ornaments
To wield old partisans, in hands as old.
Which means that not only the families and their staff are involved but almost the entire city on one side or the other.
The Prince in Romeo and Juliet is mad at the Capulet and Montague family because of their hatred towards each other . Their hatred brings violence towards the city .