The context of this quote in the story is that Emily's mother, who is home ironing, receives a call from someone--perhaps a school administrator: principal, guidance counselor or social worker--asking about Emily out of concern for her. The quote is the first line of the story.
The importance of the statement is that someone has seen a need in Emily and wants to help, so the person calls her mother. Emily's mother can understand that her daughter may need help, but does not know what she can do. She has tried so many times as Emily has been older to make up for the difficulty of Emily's first seven years of life, and has been tormented by her own concern, back and forth over the years, just as the iron moves, back and forth.
Emily's mother is speaking: she is the story's narrator.
While the narrator remembers the past, she blames herself that raising Emily was so hard for so long. The memories she brings back in recalling the past move back and forth like the movement of the iron: both the ironing and the remembering are hard, endless work for this mother. The mother compares the act of ironing to the act of raising Emily: the heat of the iron struggles to straighten the fabric of clothing ironed, straightened against its will. And with Emily, her mother has tried to straighten things out with her, too, against her will, as Emily is reserved and hard to reach.
The mother ends the conversation by asking the caller to let Emily be. She believes there is enough "bloom" or promise in Emily that she can survive in this world. If nothing else, the mother asks the caller to make sure, to make certain, that Emily knows that she is NOT like a piece of clothing to be straightened under the onslaught (the attack) of a hot iron. She is not helpless, but she can survive and succeed with the gifts that are hers alone.