The narrator and his brother Sonny have had a strained relationship for a long time. While Sonny was in high school (the narrator is 7 years older), the brothers lost their mother soon after losing their father. Sonny doesn't want to conform to his brother's wishes: staying in school in Harlem and following an employment plan. Instead, he is a musician and wants to become a jazz pianist. This mental divide, combined with the narrator's physical absence for a few years because of his military deployment, builds a wall between them.
One spring, well into their adulthood, the narrator reads in the paper that his brother has been arrested on heroin charges. He still doesn't reach out to his brother until tragedy strikes his own house. Grace, his beloved daughter, contracts polio; she suffers horribly and dies. In his own grief, the narrator begins to feel the pain of his brother.
He therefore writes the letter to his brother so that they can both find their way through the pain of their lives. The narrator also promised his mother before her death that he would look out for Sonny; after his daughter's death, he finally fulfills this promise.
Telling this story from the brother's (rather than Sonny's) point of view helps the reader to empathize with the complicated relationship that arises from loving someone while not approving of the choices that they make. The narrator is confused about how his brother could make so many bad choices, leading to a drug addiction, and the reader is able to share in this confusion. Thus, the audience gains some keen insights. The reader is also able to understand the narrator's feelings about wanting to help his brother but not knowing how to do so. In the end, he is torn about whether or not he should support his brother's musical aspirations, fearing that Sonny's jazz piano passions will only lead him back to a world of heroin and drugs.
Finally, the reader is able to feel how Sonny's performance at the end deeply moves his brother; it stirs emotions and memories within him that he hadn't imagined Sonny was capable of eliciting. If Sonny had narrated this story, the reader would have learned more about the internal struggle of battling depression and drug addictions but not how those issues affected Sonny's brother. The fact that his brother truly loves him would likely be obscured by Sonny's feeling of abandonment. Moreover, we would know that Sonny initially begged to get out of school and Harlem not because he wanted to drop out—but because he wanted to get away from drugs.