Why do Squeaky and Gretchen smile genuinely at each other after the race?

Squeaky and Gretchen smile at each other after the race because they now respect each other. Before the race, both think little of the other.
For example, Squeaky initially thinks Gretchen is vain and arrogant. Additionally, Mary Louise (formerly a close friend of Squeaky's) now socializes with Gretchen and engages in spiteful gossip against her. Rosie, the other girl who befriended Gretchen, is also on shaky ground with Squeaky because Rosie is unkind to Raymond, Squeaky's brother. When they meet on the day of the race, the girls treat each other warily, keeping each other at arms length:

Gretchen smiles, but it’s not a smile, and I’m thinking that girls never really smile at each other because they don’t know how and don’t want to know how and there’s probably no one to teach us how, cause grown-up girls don’t know either.

Squeaky and Gretchen view each other as competition. Neither thinks the other deserves to win. By the end of the race, however, both girls view each other with newfound respect and regard. Squeaky notices that Gretchen isn't just another pretty girl: she really worked hard to win.

And I lean down to catch my breath and here comes Gretchen walking back, for she’s overshot the finish line too, huffing and puffing with her hands on her hips taking it slow, breathing in steady time like a real pro and I sort of like her a little for the first time.

For her part, Gretchen now realizes Squeaky really does run well. Both girls smile to demonstrate their new regard for each other.