Ulysses, once a great hero, is very aware that he is now old. Death will be his next great journey, but he craves "something ere the end" and does not want to end his life as an "idle king" doing administrative duties, which he feels have little impact.
Ulysses has a deep sense of wanderlust—he does not want to "rest" when he still has the opportunity to "drink / life to the lees," namely, to get something out of his existence. He thinks of the things he has previously seen in his life and of his experiences in Troy, and he feels "hungry" for that kind of life, forever moving, discovering, and learning. Being still, to him, is not life, and particularly because he has so little time left, he does not want to spend it "unburnished." Ulysses seems to compare himself to a metal object that has grown dull from disuse: only by continuing to move and by being useful can he "shine."
Beautifully, Ulysses expresses that his "gray spirit" is "yearning in desire / To follow knowledge like a sinking star." He knows he has a son, Telemachus, who is now perfectly capable of looking after his lands, and he knows that he is going to die soon. Therefore, he wants to make sure he learns as much about the world as he possibly can before that moment comes. He wants to "seek a newer world" and remember the way it felt to be a young man fighting with Achilles long ago.