The narrator is fishing; this means she is either fishing for sport or for food. In either case, she is not necessarily sympathetic to this or any other fish: if fishing for sport, she is willing to put the fish through the pain of catch-and-release for her own pleasure; if for food, then she is willing to kill and eat the fish for her own survival. In either case, sympathy for the fish is unnecessary at best and detrimental at worst; too much sympathy and the narrator will starve or have little fun. However, the sight of the other hooks stuck in the fish's jaw allow the narrator a moment of intense introspection; the fish has survived and survived, only to die at her hands?

I stared and stared
and victory filled up
the little rented boat,
from the pool of bilge
where oil had spread a rainbow
around the rusted engine
to the bailer rusted orange,
the sun-cracked thwarts,
the oarlocks on their strings,
the gunnels—until everything
was rainbow, rainbow, rainbow!
And I let the fish go.
(Bishop, "The Fish," poets.org)

That she let the fish go is evidence that she felt sympathy for it at the end. Before that, she had only her own outer examination to go on; the fish is encrusted with parasites, and its skin is thin and sloughing off from age. Its eyes cannot understand her, but move in response to her movement; she, however, can understand the fish, a thing of only instinct and feeling, moving in response to stimulation. The fish has survived this long, she reasons, so who is she to kill it now? Sympathy for the fish comes from this realization, but may have been spurred by an innate sympathy for any prey animal subject to a predator such as herself.