Mama, the narrator, ultimately gives the family quilts to Maggie instead of Dee (Wangero) because she recognizes that Dee gets everything she wants, that she's even already claimed the quilts as her own, because they were promised to Maggie, and because Maggie is the daughter who wants them for the right reasons.  When Dee insists that she get to keep the quilts that she holds just out of Mama's reach, Maggie actually agrees to let her keep them, saying "'I can 'member Grandma Dee without the quilts.'"  Mama sees that Maggie is the daughter who truly understands and appreciates her family and her heritage; Dee doesn't know the stories like Maggie does, and she only want the quilts so that she can hang them on the wall. 

For Mama and Maggie, family history isn't something to be used as decoration; heritage is very much alive for them in the present: when they use handmade benches and butter churn and quilts.  Dee just wants something to show off, some proof of something in her past, not to cherish that history now.  At this point, Mama snatches the quilts from Dee and drops them into Maggie's lap, claiming the feeling is like "when [she's] in church and the spirit of God touches [her] and [she] get[s] happy and shout[s]."  She seems to have always favored Dee -- hoping for an emotional television-style reunion, raising money to send her away to school, buying her fancy clothes, and so forth -- but now she seems to really appreciate Maggie, having realized that Maggie is the one who really loves her family and does so for the right reasons.  In the end, she says, "the two of us just sat there enjoying [...]." 

The narrator does this because she knows that Maggie will make proper practical use of the quilts - the 'everyday use' of the title - whereas Maggie's sister Dee will just display them as cultural trophies. Dee has left behind her traditional family life as represented by the narrator and Maggie, and looks down on it, but makes a great show of returning to her roots when she visits. Dee comes across as flashy and superficial while Maggie is much quieter but also more sincere. Dee generally gets her own way but for once the narrator wishes to thwart her and reward Maggie instead and does so by giving Maggie the quilts.

Why does Mama give the quilts to Maggie?

In Alice Walker's short story "Everyday Use," Dee comes home to visit Mama and Maggie, who are surprised by her newfound enthusiasm for their African heritage and the ways in which she expresses it. Dee is portrayed as an outspoken, educated young woman who is attractive, entitled, and arrogant. In contrast, Maggie is Dee's quiet, passive sister who is less successful in Dee's sense of the term but possesses a genuine knowledge of their family's cultural heritage. Mama and Maggie learn that Dee has changed her name to Wangero Leewanika Kemanjo, and Dee begins to express interest in several antique items in Mama's home.

When Dee comes across two homemade quilts that are representative of the family's heritage and history, she asks Mama if she can have them. Dee believes the quilts would be impressive traditional artifacts to display, but Mama says that she promised to give them to Maggie when Maggie gets married. Dee argues that Maggie would put them to "everyday use," and Mama responds by saying that she hopes Maggie will use them often.

Even after Maggie offers to give the quilts to Dee, Mama stands up for Maggie and refuses to allow Dee to take them. Mama understands that Maggie possesses a genuine knowledge of their heritage and understands the sentiment and family meaning attached to the quilts. Mama knows that Maggie recognizes the quilts as living historical items that should be celebrated. Mama also gives the quilts to Maggie because she sees that Maggie deserves a victory in life over her sister and knows that the quilts will bring her joy.