In "The Leap," what does the author’s use of flashback add to the story?

The story is told by a narrator, not the principal character in the flashbacks.  The narrator, unlike Anna, the principal character, would not tell the story of her heroic, courageous actions as a mother the same way that a grateful daughter can and does.

She recalls her mother's life, as it relates to her, saying that her mother saved her three times, that is how the story is told through the unveiling of the three incidents.

The three flashbacks begin before the narrator was born, when Anna tried to protect her then unborn child, not the narrator, when lightening struck during her highwire act.  Anna chose to protect her unborn child, and not grab for her then husband's hand.

This led to the second flashback, when Anna went into the hospital before her baby was born, and met the narrator's father, a doctor.

The third flashback occurs when the narrator remembers her mother's courageous rescue of her from their burning house.  Anna never hesitated, even though the firemen believed that their was no rescue for the little girl in her upstairs bedroom. Anna uses her skills as a trapeze artist to swing onto the roof of the house, climb in the window and rescue her daughter.

Anna leaps out the window with her daughter held tightly against her body aiming for the fireman's net below.

Analyze the author's use of flashbacks in "The Leap".

The use of flashback in "The Leap" creates both a flow to a compelling narrative and certain unifying themes.

The title of Erdrich's story calls to mind the oft-quoted phrase of philosopher Søren Kierkegaard, who felt that "a leap of faith" was necessary for Christians if they were to accept the paradoxes that exist in Christianity. Certainly, Anna Avalon has made leaps of faith each time that she has fallen through the air with a purpose.

These leaps are what connect the narrative. With the theme of gratitude, for instance, the daughter/narrator lists three times for which she owes her life to her mother's leaps that have both given and preserved her existence. With the additional theme of "bridging gaps/making connections," the leaps made by the mother have strengthened the mother/daughter relationship. Also, the mother herself has bridged gaps in her life through marriage and motherhood. 

The chronological order of the story is as follows: 

The Flying Avalons, a trapeze act, comes to New Hampshire, and lightning strikes just as the Avalons are in midair. As she falls, Anna Avalon grabs the braided metal of a guy-wire instead of clutching her husband's ankle and falling to the ground with him. Although her hand is burnt and her arm is broken, she survives. Unfortunately, her baby is born stillborn in the hospital.

While in the hospital, Anna falls in love with her attending physician, who teaches her to read. They marry and move into an old farmhouse in the same town in which the circus accident occurred. Anna gives birth to a daughter, who is now the narrator. 

When the daughter is seven years old, the house catches fire. Anna and her husband are not at home; the babysitter awakens to find flames climbing up the stairway to the daughter's room. The sitter quickly phones the authorities, and the parents are alerted. By the time of the parents' arrival, the fire has done considerable damage. Then, when an attempt is made to reach the second story, the extension ladder breaks in half. The noise of the breakage wakes the narrator, who is around six or seven years old. She keeps her bedroom door closed and waits; soon, she looks out her window and sees her mother below.

Tearing off her dress, Anna climbs on what is left of the ladder and disappears inside the leafless branches of the tree that are near the house. She inches her way on her stomach, holding onto a tree bough that curves above another branch that touches the roof of the house. Once over the roof, Anna balances herself. Then she jumps onto the narrow branch that touches the roof, a branch that is no larger than Anna's wrist. From this branch, a standing Anna leaps onto the new gutter that hangs over her daughter's bedroom. Hanging by the backs of her heels, the mother smiles as she taps on the window of her daughter's room. When the window is opened, Anna swings down and enters the room. Holding her daughter, Anna signals to the rescuers, and they "flew out the window, toward earth. . . and the painted target of the fire fighters' net."

The narrator, who is the daughter of Anna, explains that she owes her life to her mother three times. She then proceeds to narrate the tale of Anna Avalon and the three incidents which involve her. At the end of her narration, the daughter reflects that her mother has been right to tell her that as a person falls, there is time to think. During this time for thought, Anna Avalon saved her own life and, later, her daughter's life.