In the story, John Byro is an Assyrian farmer. He is described as lonely. His loneliness is one reason why he visits the narrator's house. He is described as a visitor, rather than a stranger, and the narrator knows a little bit about him, including the fact that he learned Armenian to alleviate his loneliness, so the impression is that he is perhaps a friend or acquaintance of the narrator's family. He, therefore, perhaps visits the house in that capacity. Later in the story, John Byro calls the narrator and his cousin the "sons of my friends," confirming this impression.
Another reason why he visits the narrator's house is to perhaps spread the news that his horse has been stolen. He also maybe hopes to elicit sympathy from he narrator's household, as, without a horse, his surrey (carriage) is useless.
John Byro seems to be upset, claiming that he has "walked ten miles" to get to the house, and "slam[s] the screen door" when the narrator's uncle doesn't give him the sympathy he wants. The narrator's mother explains to the uncle that "he (John Byro) has a gentle heart" and that he is upset is because "he is homesick." This suggests that John Byro visits the narrator's house for the company and for sympathy because of his stolen horse.