In 1961, the financially strapped author Pamela "P. L." Travers reluctantly travels from her home in London to Los Angeles to work with Walt Disney at the urging of her agent, Diarmuid Russell. Disney has pursued the film rights to her Mary Poppins stories for twenty years, having promised his daughters that he would produce a film based on them. Travers has steadfastly resisted Disney's efforts because she fears what he would do to her character. However, she has not written anything in a while and her book royalties have dwindled to nothing, so she risks losing her house. Still, Russell has to remind her that Disney has agreed to two major stipulationsâno animation and unprecedented script approvalâbefore she agrees to go.
Travers' difficult childhood in Allora, Queensland, Australia, is depicted through flashbacks, and is the inspiration for much of Mary Poppins. Travers idolized her loving, imaginative father, Travers Robert Goff, but his chronic alcoholism resulted in his repeated firings, strained her parents' marriage, and caused her distressed mother to attempt suicide. Goff died at an early age from tuberculosis when Travers was seven years old.
In Los Angeles, Travers is irritated by what she perceives as the city's unreality and the inhabitants' intrusive friendliness, personified by her limousine driver, Ralph. At the Walt Disney Studios in Burbank, Travers meets the creative team that are developing Mary Poppins for the screen: screenwriter Don DaGradi, and music composers Richard and Robert Sherman. She finds their presumptions and casual manners highly improper, a view she also holds of the jocular Disney.
Travers' working relationship with Disney and his team is difficult from the outset, with her insistence that Mary Poppins is the enemy of sentiment and whimsy. Disney and his people are puzzled by Travers' disdain for fantasy, given the nature of the Mary Poppins story, as well as Travers' own rich imagination. She particularly objects to how the character George Banks, the estranged father of the children in Mary Poppins' charge, is depicted, insisting that he is neither cold nor cruel. Gradually, they grasp how deeply personal the Mary Poppins stories are to her and how many of the characters were inspired by her past.
The team realize Travers has valid criticisms and make changes, though she becomes increasingly disengaged as painful childhood memories resurface. Seeking to understand what troubles her, Disney invites Travers to Disneyland, which, along with her developing friendship with Ralph, the creative team's revisions to the George Banks character, and the addition of a new song and a different ending, help dissolve Travers' opposition. Her creativity reawakens, and she begins working with the team; however, when Travers discovers that there is to be an animation sequence, she confronts Disney over his broken promise and returns home.
Disney learns that Travers is actually her pen name, taken from her father's given name. Her real name is Helen Goff, and she's actually Australian, not British. This gives Disney new insight into Travers, and he follows her to London. Arriving unexpectedly at her door, Disney tells her that he also had a less-than-ideal childhood, but stresses the healing value of his art. He urges Travers to not let deeply-rooted past disappointments dictate the present. Travers relents and grants Disney the film rights.
Three years later, in 1964, Mary Poppins is to have its world premiere at Grauman's Chinese Theatre in Hollywood. Disney has not invited Travers, fearing how she might react with the press watching. Prompted by Russell, Travers shows up unannounced at Disney's office; he reluctantly issues her an invitation. Initially, she watches Mary Poppins with a lack of enthusiasm, particularly during the animated sequences. She gradually warms to the rest of the film; however, becoming deeply moved by the depiction of George Banks' personal crisis and redemption.

Question: How does Walt Disney gain Traver's trust again?
Answer: Disney travels to London to persuade Travers to continue working on her project. 
Alternative answer: He talks about his own childhood