This very interesting question refers to Book 21 of Homer's Odyssey. In this book, Odysseus' wife, Penelope, has proposed a contest in which the person who can string her husband's bow and shoot an arrow through a series of axe-rings will become her husband.
At Odyssey 21.80-135, Telemachus sets up the axes and then has a try at stringing the bow. He probably would have succeeded if he had not seen his father giving him a signal to stop his attempt.
When Telemachus takes the bow, he says that if he can accomplish the task, then
I shan’t be so upset by my dear mother’s departure for another house, seeing I myself will be a man capable of winning fine prizes like my father. (A.S. Kline translation)
So, from this comment, it would appear that Telemachus is motivated by a number of things. First, stringing the bow would serve as a sort of rite of passage for himself. He wants to prove to himself that he is capable of accomplishing the sorts of deeds that will allow him to gain glory. Second, I would imagine that sheer curiosity about the bow motivates him to try to string it. He may want to find out if he is as mighty a man as his father.
At any rate, this is a very curious moment in the epic as Telemachus, at least on the surface, appears to be on the verge of becoming another Oedipus.