The meaning here is that the fish will eventually die of the wound suffered from the harpoon that Santiago has struck it with. Additionally, the harpoon is here represented symbolically.
The harpoon is a symbol and metaphor for the death of the fish. As the harpoon-as-death metaphor goes, the harpoon is in the fish so death is in the fish too.
The line, "Then the fish came alive, with his death in him..." describes the response that the fish has to being struck. For a while, he still lives, but eventually the wound made by the harpoon will kill him.
It may be helpful to consider an analogy for this situation. If a person contracts an un-treatable terminal disease or deadly virus, we might say that the person "has their death in them" because the disease/virus is already present and will inevitably cause death, though that death will not be instantaneous.
The writhing of the fish is a physical response to such a state of being, aware of the presence of the thing that will kill it and rebelling against the inevitable. This response shows some of the strength of the fish and some of its nobility, but the event depicted here functions also as a fulfillment of Santiago's promise:
“Fish, I love you and respect you very much. But I will kill you dead before this day ends.”