Romeo is motivated to fight Tybalt because Tybalt killed his best friend Mercutio. After Mercutio died a rage came unto Romeo and they started to fight. Of course Romeo won. read more
But Romeo did not want to fight him, because Tybalt was related to Juliet. Mercutio then stepped in and challenged Tybalt to a duel, which Tybalt accepted. There was no reason for these two to be fighting except that they were quarrelsome. read more