Explain how Anne's character undergoes a wonderful and significant transformation in the Annex.

Anne's development in her diary is extraordinary. She becomes more of a young adult through her trials in the Annex, growing both as a writer and a human being.
Anne begins her diary talking about the mundane things in her life—friends, boys, school, people she dislikes, her birthday party—with only a few mentions of Jewish persecution (having to wear the yellow star, curfews, not being allowed to have bikes). Once the family moves into the Annex, Anne becomes more thoughtful, even philosophical, worrying about more than just her own physical comforts.
Anne writes about the events outside more frequently, wondering about the outcome of the war and the presence of anti-semitism in Europe. She thinks about religion and love. She even thinks about women's place in the world due to sexist attitudes of the period, and how she wants to have both a family and a career. Most teenagers begin to become more independent in their mid-teens, thinking about topics and forming their own views on them, which Anne does a great deal in her diary.
Of course, she is still a teenager. She retains an uncharitable, even petty, view of her mother. She gets sullen when the adults treat her unfairly. But her growth is nevertheless impressive in such a short span of time.
In the end, Anne goes from being a perky child to a thoughtful, optimistic young woman who certainly had potential to become an engaging, humane writer had she only gotten to see the end of the war.

Anne experiences substantial personal growth throughout the Diary, developing considerable maturity during her claustrophobic existence in the Secret Annex. As well as having to deal with the normal problems of adolescence, she somehow has to stay strong in the face of imminent danger. As a Jew in Nazi-occupied Amsterdam, Anne knows that she and her family can be arrested and sent to a concentration camp at any moment. This would be a terrifying experience for anyone, especially a young adult with limited experience of the world. Yet as well as displaying considerable maturity for someone of her age, Anne also seems to dig deep and find the necessary strength to help her get through such a dangerous, life-threatening situation. And toward the end of her diary, Anne attains a broad perspective on world events that is not just hopeful and optimistic, but also incredibly humane.