How do Santiago and the sea turtles symbolize each other in The Old Man and the Sea?

Santiago feels a kinship with the turtles because his hands and feet are like theirs, horny and hard. The turtles are tough and impervious to the stings of the treacherous jellyfish, just as Santiago is to the stings of the sea (and more largely to the stings of nature). Like the turtles, he won't give up. He and the armored turtles are part of the pre-ordained order of things, made of the same stuff. The turtles do what they have to do simply because, existentially, that is what they were made to do, and so does Santiago. Like Santiago, the turtles are real and solid, and they challenge what is false, represented by the stinging jellyfish:

The iridescent bubbles were beautiful. But they were the falsest thing in the sea and the old man loved to see the big sea turtles eating them. The turtles saw them, approached them from the front, then shut their eyes so they were completely carapaced and ate them filaments and all. The old man loved to see the turtles eat them and he loved to walk on them on the beach after a storm and hear them pop when he stepped on them with the horny soles of his feet.

Santiago also says he has a heart like the turtle's, that will "beat for hours after he has been cut up and butchered." Both the turtle and Santiago symbolize what is hard and impenetrable on the outside but pulses beneath the surface with hidden, stubborn life (hearts that won't stop beating).

Compare and contrast Santiago with the sea turtles.

While at sea on the second day of Hemingway's narrative, Santiago sees the respected sea turtles who venture forth bravely to eat the poisonous jellyfish.  Closing their eyes so that they are carapaced, the heroic turtles eat the jellyfish filaments and all, bravely doing what they must to eat.  Likewise, the determined Santiago goes forth onto his beloved sea, la mer, facing whatever dangers he must in order to eat. For, the sea feeds him, as well. 

As he admires the turtles, despite his having worked on turtle boats for many years, Santiago feels a sympathy for them, understanding that they, like him, venture forth and seek food dangerously because they must.  Yet, he is unlike them because they are truly a part of the sea while he is only at sea in a boat.  Yet, they have been cut up by man, as he has and his heart has had to beat outside his body figuratively just as the turtle's will do so in reality when they are captured and killed for food.  Watching the turtles seek their prey, Santiago thinks,

Most people are heartless about turtles because a turtle's heart will beat for hours after he has been cut up and butchered.  But...I have such a heart too and my feet and hands are like theirs.

Santiago's hands are rough and cut by the sea; like the turtles he is steady and persistent in going after his catch.  And, like the turtles, his heart must sometimes beat "outside" his body when he fails to catch anything, or when others taunt him.  Although old, Santiago is hardy like the old loggerhead turtles; still, he is not a real part of the sea as are the eternal turtles.

Why did Santiago love turtles in "The Old Man and the Sea"?

Santiago despises the Portuguese man o' war. The turtles eat these jellyfish, "filaments and all," and this is one of the main reasons he likes the turtles. As an older man who still prides himself on his strength and endurance, he feels a kinship with and has respect for these turtles because they can eat the very poisonous Portuguese man o' war without being injured or poisoned.

Santiago respects the marlin he battles with. It is a graceful and powerful animal and he recognizes this. Likewise, he admires the green turtles and hawk-bills for their "elegance and speed and their great value." It seems that Santiago also admires the turtles because they can live to a ripe old age. As an old man himself, they probably give him comfort that a fellow creature can live a long life and still be effective. Just as Santiago respects and relates to the marlin, he has a similar kinship with the turtles. He notes how their hearts still beat after they're killed and how their weathered hands and feet resemble his own aged hands:

But the old man thought, I have such a heart too and my feet and hands are like theirs.

How does Santiago relate to the turtles in The Old Man and the Sea?

Santiago, the old man, thinks about how he loves turtles. He loves the various kinds of them: the elegant green turtles and hawk-bills for their speed and "great value," but also the awkward loggerhead species.

He feels, as well, a strong sense of identification with the turtles. He likes them because they eat the jellyfish, which he calls the "falsest" creatures in the sea. He hates the jellyfish because of the welts and sores they leave on his arms when they get caught in the fish lines. But he enjoys watching the turtles eat them:

The turtles saw them, approached them from the front, then shut their eyes so they were completely carapaced and ate them filaments and all.

The old man's descriptions of the various kinds of turtles shows he knows them well from his many years at sea, but he says he also feels sorry for them, even the great turtles that are as long as his boat and weigh a ton. He feels for them because their hearts will go on beating for hours after their bodies have been cut into pieces.

However, their powerful hearts are also why the old man identifies with them. He muses that he too has a heart that doesn't give up easily:

But the old man thought, I have such a heart too....

In addition, he identifies with the turtles because they are vulnerable to ruthless shark attacks. The old man notes that the sharks will attack the turtles when they are asleep and bite off their legs and flippers.