How does "A White Heron" exemplify local color realism?

Jewett's story is a great example of "local color," a term that refers to literary fiction primarily about a specific place or region. The story exemplifies this is several ways.

Setting:

The story is set in rural New England, close to the coast. While the exact location is never explained, the woods and marshes around Sylvy's house serve as a kind of character in their own right. The opening of the story, in which Sylvy goes looking for a wayward cow, is given primarily to description of this place—which has transformed Sylvy, who had lived before in a town. The narrator writes that it seemed she "never had been alive at all" before she came to the farm.

Characters:

Sylvy and her grandmother live lives that are formed by this remote setting. Their rustic cabin and Sylvia's life among the creatures of the forest almost suggest something out of a fairy tale. The bird hunter, as a representative of the outside world, throws their quaint ways into relief. It is also characteristic of local color that the great pine tree and the heron itself both function as characters too. While these things are not directly personified, they do form part of community that includes Sylvy and her grandmother.

Dialect:

Jewett goes to some trouble to represent the New England dialect the grandmother speaks in her prose. Her unique way of speaking marks her out as belonging to this place, just as the hunter's standard English marks him as an outsider.

"A White Heron," by Sarah Orne Jewett, is an excellent example of local color because of its realism. Local color refers to an author's use of the language (dialect) and manners of a particular group of people.

In this story, Jewett uses dialogue to show the difference between the "locals" and outsiders. Consider the differences between the following statements:

"Put me anywhere you like," he said. "I must be off early in the morning, before day; but I am very hungry, indeed."

 "Dear sakes, yes," responded the hostess, whose long slumbering hospitality seemed to be easily awakened. "You might fare better if you went out to the main road a mile or so, but you're welcome to what we've got. I'll milk right off, and you make yourself at home. 

The first statement is spoken by the ornithologist who is searching for the white heron. His language is more refined and he does not use contractions, in contrast to Sylvia's grandmother's slang ("dear sakes") and her contractions. He is not formally part of the same region (culture, lifestyle, manners) as Sylvia and her grandmother, which is why he knows his offer of ten dollars (so little to him) will be a great enticement to this poor family.

Jewett also adds local color by her use of conversational topics and practices common to the region in which Sylvia and her grandmother live.

Soon it would be berry-time, and Sylvia was a great help at picking. The cow was a good milker, though a plaguy thing to keep track of, the hostess gossiped frankly.

These two lines evoke the image of a farm-woman who is sitting down at the table for a quick gossip before beginning her homely chores once again; she spends her time thinking about picking berries during "berry-time" and fretting about the sporadic milk-giving habits of her cow. 

Both the dialogue and the use of details in this story mark it as particular to a time and place, which is what the term "local color" means. Your question adds the idea of realism, but I wonder if you meant regionalism, as the same elements which signify local color are also indicators of regionalism. In any case, these details make the story believable and realistic, and this story is a good example of regionalism, as well. 

Sarah Orne Jewett's "The White Heron" is an example of literary Realism. Realism was a reaction to Romanticism. Realists wished to portray life as it was, unlike the Romantics, who preferred to show the world as they wanted it to be—looking at the world through "rose-colored glasses." Realists wanted to show the world without artificiality, focusing more on events and characters who were probable and realistic.

Jewett's short story opens realistically:

The woods were already filled with shadows one June evening, just before eight o'clock, though a bright sunset still glimmered faintly among the trunks of the trees. A little girl was driving home her cow, a plodding, dilatory, provoking creature in her behavior, but a valued companion for all that. They were going away from whatever light there was, and striking deep into the woods, but their feet were familiar with the path, and it was no matter whether their eyes could see it or not.

Here, the text depicts a typical evening in the woods. The sun is setting, and a little girl is walking home with her cow. The path, walked by uncountable steps, is worn and known.

As the young girl (Sylvia) walks home, afraid her grandmother will be angry at their lateness, she comes across a young man. The young man questions the whereabouts of a white heron. Again, this seems very realistic and plausible. The two are in the woods, and they happen to come across one another. The young man strikes up a conversation with Sylvia about the bird. He even offers ten dollars to anyone who can lead him in the right direction to find the bird.

In the end, Sylvia considers the money and her own feelings about the bird. She refuses to tell the young man about the location of the bird; she simply cannot bring herself to disclose the location knowing the bird could be harmed.

This aspect of the story is realistic because it makes readers think about their own morality and worth. Jewett's point, in making the story realistic, is to provide readers with a situation in which they could see themselves. The readers, like Sylvia, would need to make a decision about disclosing the location of the heron or keeping the heron safe.

The setting and characters are realistic, ones that readers can picture and (perhaps) relate to. Jewett includes no supernatural aspects, no exotic settings, and no "larger than life" characters. The story is one which provides an observer's point of view, where neither the author nor the reader can intervene.

An author of a work of literary realism aims to depict the everyday world in a realistic -- as opposed to idealistic -- way.  This is what Jewett attempts to do through her description of the banal, commonplace activities of life on a farm: Sylvia's nighttime, muddy search for Mistress Moolly at the story's beginning, cow-milking, berry-picking, toad-watching, and keeping company with chickens.  Nothing is idealized or romanticized; the effect of nature on Sylvia is immense, to be sure, but there are no descriptions of, for example, some blissfulness that nature imparts to the child.  Rather, Jewett simply talks about Sylvia as though she were a part of the nature she so loves, comparing her throughout the story to a flower, a bird, even a star. 

Further, the opposition Jewett establishes between Sylvia, a representative of nature, and the stranger, representative of "civilization," helps to establish the story's realism.  She depicts the steady encroachment of the city on the wild and the damage to nature that such trespass inflicts.  The stranger nearly corrupts the innocent Sylvia into betraying the heron by offering her money and gifts, by flattering and charming her.  He isn't evil, but it seems as though he does not understand the danger he, and his way of life, pose to her and her woods.  By the end, after Sylvia has decided not to tell him where the heron is, and he leaves, she thinks of "the piteous sight of thrushes and sparrows dropping silent to the ground, their songs hushed and their pretty feathers stained and wet with blood."  Such a true-to-life, matter-of-fact description of the stranger's way of appreciating nature helps to further mark this text as a work of realism.

What aspects of realism are evident in Jewett's "A White Heron"?

Realism is a movement in literature in which authors attempt to depict life realistically. This means that authors tend to describe the minutiae of daily life in detail and tend to feature characters who are average and ordinary. Unlike works of romanticism, realist works are not very dramatic and do not center around crisis moments as plot points. Regionalism (also called local color fiction) is a more specific subgenre of realism in which authors capture a realistic vision of life in a particular area of the country.

In Sarah Orne Jewett's "A White Heron," the landscape and culture of the rural northeastern United States is portrayed. As in most regional works, Jewett's narrator begins with a detailed physical description of the setting. The main character, Sylvia, is also introduced, and she is immediately connected with the natural setting that surrounds her. As the narrator explains, "it was her greatest pleasure to hide herself away among the huckleberry bushes" (paragraph 2) and to go out and find the cow that had wandered into the pasture. Local color is also introduced into the story through dialogue, and when Mrs. Tilley and the sportsman speak in the story, we can understand speech patterns and concerns of people in the region. Later in the story, Sylvia must resolve a conflict in which she can either tell the hunter where to find the white heron or keep the information to herself. Sylvia decides to do the latter, aligning herself with her natural environment. The story is an example of regionalism because it showcases the landscape and the people of the setting, but Jewett takes this a step further in actually connecting the protagonist to the region on a deeper level due to her emotional bond with the heron.