The heron symbolizes nature and the damage that society was doing to nature in the late 19th century. At that time, people began to develop an interest in beautiful birds especially, and women would wear stuffed birds and extravagant feathers in their statement hats. "Plume hunting" referred to the hunting of wild birds to harvest their feathers, typically for use in fashion. The hunter in the story represents this encroachment of society.
One clue is in Sylvia's name, the root of which is "sylvan," a word that means relating to or characteristic of the woods or forest. At various points in the story, Sylvia is connected or compared to various aspects of nature: a flower, a bird, a star. And she comes to understand the respect that she has for nature through the figure of the bird; she knows its value is worth more than money. The narrator says,
The murmur of the pine's green branches is in her ears, she remembers how the white heron came flying through the golden air and how they watched the sea and the morning together, and Sylvia cannot speak [to the hunter]; she cannot tell the heron's secret and give its life away.
Sylvia understands that it is counterintuitive to say that one appreciates and loves nature, and so one kills it in order to appreciate it all the more. She mourns the birds the hunter shoots because, even if their feathers remain physically pretty, what really made them beautiful was their song and their life, which are now gone. Sylvia would not, could not, sell the heron, just as she could not give up her life in the forest. The forest, the heron, even Sylvia, are all one: they are Nature -- innocent and lovely and corruptible by humans.