Montresor begins the story by saying, "The thousand injuries of Fortunato I had borne as I best could; but when he ventured upon insult, I vowed revenge." Montresor employs a figure of speech called hyperbole, also referred to as overstatement, to emphasize just how incredibly offended and insulted he felt by Fortunato. It is unlikely that Fortunato had actually injured him a "thousand" times, but Montresor felt as though he had. This, in part, motivates his need for revenge.
Further, Montresor says that his family motto is "Nemo me impune lacessit," which translates to "You will not harm me with impunity." This means that no one can harm a Montresor without being punished or repaid for it. Thus, if Montresor believes that he has been harmed by Fortunato, his family pride will not permit him to allow it to go unavenged. He clearly possesses such a sense of family pride, as he states that "The Montresors were a great and numerous family."
Therefore, what motivates him to him to seek revenge on Fortunato is a mixture of wounded personal pride and family honor.