Odysseus cares because the cyclops asked him his name before and he replied "nobody" to which was used strategically to make sure he had no help when he called for help and said nobody was attacking him. So when he escapes basically defeating the cyclops,He basically brags that he did such a feat by revealing his True name. This however tends to backfire as since he revealed his name, the cyclops curses him with Poseidon since he's also Poseidon son Polyphemus. This causes Odysseus to basically lose his crew to the sea and struggle to get back home to Greece.
