How does Raymond's run contribute to the resolution of the story?

Raymond's having run alongside her on the other side of the fence reminds Squeaky of what she has revealed in the exposition as her duty to her family:

All I have to do in life is mind my brother Raymond....

Now, she "minds" her brother more closely and perceives potential in her brother that she has not noticed before; and, having done so, Squeaky abandons her preoccupation with herself, 

[A]nd I look over at Gretchen.... And I smile. Cause she’s good, no doubt about it....And she nods to congratulate me and then she smiles. And I smile. We stand there with this big smile of respect between us.

Squeaky now has learned to find merits in others. As she acknowledges Raymond's talent, so, too, does she acknowledge that Gretchen is a real person on her own, who does not act as though she is better than someone else; nor does Gretchen have any personal motives for pretense. She is simply pleased and proud of Raymond's efforts, as well as those of Squeaky. Her genuine smile and gestures toward Squeaky convey her sincere feelings, feelings that Squeaky realizes are much the same as those she has herself toward Raymond.