A first-person perspective is very useful for giving us an insight into a character's mindset. In "Raymond's Run," the first-person narrative allows us to put ourselves in the shoes of Squeaky, a young girl from Harlem who's just about the best runner around.
As most readers of the story are not athletes, let alone successful ones, the first-person narrative of the story gives us a fascinating glimpse of a world to which most of us don't belong. We can also learn how Squeaky feels about herself. She tells us that she's the fastest thing on two legs, not something we'd be likely to hear from anyone else.
In fact, as she tends to be surrounded by mean girls who give her disabled brother, Raymond, all kinds of smack talk, it's a fair assumption that such a description could only come from herself.
The cruel behavior of these girls adds to the sense that it's Squeaky and Raymond versus the world, and the first-person narrative adds to the sense that brother and sister are somehow in a world of their own, separate and distinct from everyone else.
The first-person narrative also offers up a kind of tunnel vision, which is appropriate to Squeaky. When she turns up at the May Day celebration, Squeaky isn't in the least bit interested in dancing round the maypole; she just wants to run—and win—the big race. Her feet are for running, not dancing.