Works of literature such as The Old Man and the Sea are not meant to be taken purely at face value. The reader is challenged to read between the lines, drawing parallels between the story and life. The Old Man and the Sea is, on the surface, actually a pretty banal, mundane story. Not much happens, really, and none of it is especially interesting. But the entire arc of the story is symbolic of a greater life struggle. Let's face it: a really big fish does not make for an enduring work of literature. A really big fish symbolic of the human impulse to seek great things even though attaining them would mean our own destruction.... Well, you get the idea.
At the end of The Old Man and the Sea, the old man's failure is final and complete. Life has nothing left for him, and he has nothing left for life. At face value, he goes to sleep. But when one considers the symbolic meaning of the story, his sleep can only mean his death. It is the natural resolution to all that has come before. Sleep is an oft-used reasonably well understood symbol for death, and this case is no exception. Hence our claim that the old man "DIES."