What draws Prince Escalus into the story?
The first time he enters the story, it is to speak sternly about the fighting between the two families. He makes the fighting boys throw their weapons on the ground. The second time he enters the play it is to investigate the deaths of Mercutio and Tybalt. He banishes Romeo. Finally, he enters the scene a third time to investigate the deaths of Romeo and Juliet. The Prince finally gets the Peace that he was been wanting between the two families, but at a heavy cost, the lives of Romeo and Juliet. He says as much when he says, "a glooming peace this morning with it brings" (V.iii.304-5).
Prince Escalus is drawn into the story when the feud between the Capulets and Montagues ensues in the courtyard endangering the lives of many innocent bystanders. He lets them know that they have disturbed the peace 3 times and if they do it again their lives will pay for the fight. He meets with Lord Capulet and Lord Montague separately to try to settle the fighting. Prince Escalus is the "law enforcement officer."
Prince Escalus enters the story in the very beginning when the Capulets and Montagues are feuding in the town square. Prince Escalus is like the "chief of police" and warns both families that this will not be tolerated and their lives will pay for the price of peace if they disturb the streets again. He is in the beginning, middle, and end of the play.