The narrator of “Death in the Woods” states that it was the woman’s destiny to “feed animal life.” Do you feel that this quote adequately explains the significance of the story? What are some of the other ideas and themes present?

No, the quote that the woman was destined to feed animal life does not adequately explain the significance of the story.
First, the story is about more than merely telling the old woman's story. It is about how, in general, we tell any story. The narrator doesn't simply provide a straightforward start-to-finish narrative but consciously explains how the story has come back to him and notes where he remembers new things. He repeats himself and circles back over his narration. He shows us how a narrative emerges from fragments of memories. He emphasizes the fictive nature of the story in telling us that he has learned much of it second-hand, pieced together from what others have told him.
Further, the story is also about the teller. It is not only the woman's story, but his story too. This seems to imply that every story we tell is in some way our own story. A story means something to us and is important enough to tell when it relates to our truth.
Finally, it's important to note the woman's fate as an animal feeder comes from her hard life—the people around her have acted like animals to her and treated her like an animal. The story of the animals and her struggle to keep them alive is also the story of her own struggle to survive.