Romeo becomes enraged at Tybalt because he slayed Mercutio, one of his closest friends.
Romeo pursuits Tybalt, with Benvolio following him in attempt to stop him. Fueled by his own fury, Romeo is wishing to slay Tybalt because of his nefarious deeds.
Benvolio finally manages to settle Romeo, and begins to inform him of the wretched consequences he would receive if he were to kill Tybalt.
Romeo, no longer consumed by rage, realizes what would happen from his wrongdoings, and ends his pursuit of Tybalt.
As a result of abandoning his pursuit of Tybalt, Romeo will not be banished from Verona. Therefore, Romeo will not end up dying because there will be no cause for Friar Lawrence to devise a plan to have Romeo and Juliet leave Verona together because he never would have left Verona in the first place.