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Tybalt comes looking for Romeo because he wants to challenge and kill him. Tybalt wants revenge since Romeo showed up at the Capulet Party. What is Romeo's reaction when Frair Lawrence tells him Escalus has banished him? He tells the friar that it is worse than death.
Besides, what happens when Tybalt come looking for Romeo?
He wants to challenge Romeo to a duel. Romeo interferes with the fight and Tybalt reaches around Romeo and stabs Mercutio because Mercutio couldn't see Tybalt.
Then, why does Tybalt look Romeo?
In Act III, Scene 1 of Romeo and Juliet, Tybalt is looking for Romeo to fight him for attending the Capulet ball uninvited. Because the Capulets and Montagues are in a family feud, Romeo “crashing” the party is disrespectful and forbidden. Because of this, Romeo doesn't want to fight Tybalt, but he can't tell him why.
Romeo protests that he has good reason to love Tybalt, and does not wish to fight him. He asks that until Tybalt knows the reason for this love, he put aside his sword. Mercutio angrily draws his sword and declares with biting wit that if Romeo will not fight Tybalt, he will. They fight, and Romeo kills Tybalt.