Emily is bragging, though what she says is true. This statement provides us with a concise exposition of Emily's character and also connects later to her realization that people are essentially blind.
Emily is very bright. Though she is bragging about being the "brightest girl in school", she is not exaggerating.
She is bright, articulate, and, despite the anxiety she shares with her mother, a beautiful creature.
Other characters recognize Emily's intellect as well. George comments on Emily's ease with school work in their first meeting of the play.
When Emily tells her mother that she is the brightest girl in her class, she strikes a tone of childish confidence. She is literally expressing the brightness of her character, her hopes, her joy, and her desire for approval and success.
This tone can be seen as "closed off" in the context of Emily's post-death revelation. As a girl, Emily is aware only of herself and her needs and is blind to the wealth of beauty around her.