Santiago has not caught a fish in eighty-four days. He is very old, but he is determined to continue to try despite his old age and/or poor luck. There is the implication that Santiago is too old, that he's lost the mental and physical ability to continue to be a successful fisherman. Even Manolin's parents have told him to stop fishing with Santiago, claiming he is "salao" - unlucky. Santiago wants to prove to others and to himself that he is still useful. When he begins his battle with the marlin, it becomes apparent that he respects the fish. When he first hooks the fish, he says, "I love you and respect you very much. But I will kill you dead before this day ends." Santiago thinks the fish is noble. Santiago concludes that in killing the marlin, he will earn the marlin's respect as well. Since he respects the marlin so much, it will be a noble battle. In this sense, Santiago empathizes with the marlin; he feels a camaraderie with the marlin. Both he and the fish are struggling nobly against each other. Santiago wants to prove to the marlin that he is worthy of catching and killing such a great fish (just as he wants to prove it to himself and others.)
As the struggle continues, Santiago feels sorry for the marlin. And his respect for the noble fish increases. He considers who will eat the marlin:
How many people will he feed, he thought. But are they worthy to eat him? No, of course not. There is no one worthy of eating him from the manner of his behaviour and his great dignity.
The reason Santiago becomes so infatuated with proving himself to the marlin is because he respects the marlin so much. To prove himself to one (the marlin) whom he holds in the highest respect is a great accomplishment. In fact, Santiago might feel more satisfaction in proving himself to the marlin than he would in proving himself to his fellow fishermen.