What was Squeaky's important lesson that she learns from her brother?

“Raymond’s Run” tells the story of Hazel Elizabeth Deborah Parker, a young girl growing up in Harlem who goes by the name of Squeaky. The story focuses on Squeaky’s preparation for the May Day track race and her relationship with her brother and the other girls at her school. Although Squeaky’s focus is on winning the race, she learns an important lesson from her brother about the value of forming authentic relationships with those around her. She realizes that her previous relationships—many of which are antagonistic in nature—have been negatively constructed, and it is particularly important for girls to help one another. 
When the race starts, Squeaky realizes that her brother Raymond is running parallel to her on the other side of the fence. It is the first time that she can recall seeing him run, and she is struck by the strength of his stride and his technique. Although her focus throughout the story has been on proving that she is the fastest, in this moment her thoughts shift, and she realizes that it might be more important to help him. Although it is not defined throughout the story, it seems clear that Raymond may have a developmental disorder: 

And I have a big rep as the baddest thing around. And I’ve got a roomful of ribbons and medals and awards. But what has Raymond got to call his own?

Squeaky wins the race, but her thoughts drift to those around her, moving first to how she can help her brother and then to Gretchen. Before the race begins, Squeaky’s teacher suggests to her that she might throw the race to make Gretchen, the new girl in school, feel better about herself; while this idea is initially laughable, after Raymond’s race, she is not only willing to give up her place in the spotlight, but she also sees how her emphasis on competition has caused her to fight with the other girls around her. The main external conflict in the story is not the race but a chance encounter with Gretchen and her friends. Later, Squeaky realizes that this conflict is based not on Gretchen but rather on the way that society has shaped the two girls:

We stand there with this big smile of respect between us. It’s about as real a smile as girls can do for each other, considering we don’t practice real smiling every day, you know, cause maybe we too busy being flowers or fairies or strawberries instead of something honest and worthy of respect . . . you know . . . like being people.

The fake smiles of their first encounter are replaced with smiles of mutual respect and admiration. Just as the narrator rejects being an object on a stage, she too rejects the forced antagonism that the track meet has produced.