In a nutshell, Anne and Peter's relationship changes in these scenes from them being acquaintances to close friends and, at least from Anne's side, a little more than that.

Towards the end of scene 1, Anne remarks to Peter that it's surprising that it has taken them almost a year and a half to start talking to each other. It is after Anne storms off into her room after one of her habitual fights with her mother, Mrs. Frank, that Peter decides to follow her. He brings her the cake that she had left behind, and this is when the two have their first real talk, sharing commiserations about life in the annex. In the spirit of a true friend, Peter tells Anne that she can come to his room any time she feels as though she needs to blow off steam.

In scene 2, we see further evidence of how Anne and Peter's relationship has become closer. She is going to meet him for the closest thing to a date that the annex can offer—time spent alone with Peter in his room, away from the prying eyes of the other residents. During their conversation, Anne seems to come to the realization that Peter likes Margot better than her, but this does not stop her from hoping for a goodnight kiss.

In the play, just as in the book version of Anne’s diary, Anne grows closer to Peter. He becomes her only meaningful ally other than her father and her diary, “Kitty.” In act 2, scenes 1 and 2, we see Peter come to Anne’s defense and the dialogue that reveals their growing friendship.

When the act opens, Peter’s parents have an argument which ostensibly is about Mrs. Van Daan’s fur coat but in reality is about much more than that. They argue about her attachment to the coat and her inability to let go of it so that her husband can sell it to raise badly needed funds to keep the family going. From there, the argument takes a turn and reveals the underlying source of friction between them, which is that Mrs. Van Daan refused to flee Amsterdam when Hitler first came to power and they had a chance to escape. Anne tries to buffer the argument and Mrs. Van Daan turns on her and insults her, saying that Anne speaks too freely and should keep out of this discussion.

Hurt, Anne leaves, and Peter goes after her to apologize for his parents. This draws the two closer together and shows their budding friendship. Peter laments about his parents’ behavior and says that he hopes that he does not turn out like them when he is older. He then tells Anne that she is always a big help to him because her cheerful demeanor keeps him from being depressed.

Anne reveals that she does not feel cheerful deep inside. Anne acknowledges that they share a bond that they never realized before. Even though they have lived in the annex together for a year and a half, “this is the first time we’ve really talked,” she says. Unlike her mother, Mrs. Van Daan, and even sometimes her sister, Peter compliments Anne and is there to listen to her when she is worried or angry.

In the play The Diary of Anne Frank, Anne and Peter's relationship changes in act 2, scenes 1 and 2 because tension in the annex is growing among the adults, and the fact that the children are also maturing. For example, after Peter accidentally smashes a light that scares a burglar away, Mr. Kraler announces in scene 1 that a workman wants blackmail money not to report them. Dussel blames Peter, which starts an even bigger argument, and soon everyone is yelling. As a result, Anne blows up at the adults and storms off. Peter takes her the piece of cake she left behind, and this is when they first confide in each other about their struggles living in the annex with such stressed adults. 

By the second scene, Anne is shown trying on clothes with Margot and getting ready for her first date with Peter. Then, during their meeting in Peter's room, not only does the couple talk about kissing and proper courtship, but they also talk about friends and having seen each other at school before going into hiding. They discuss how each one felt about the other when they first met compared to becoming interested in each other, now. They realize how much they have matured while in hiding. This date helps to close the chapter on their past, in a way, as they approach a more intimate relationship at the present. Happily for Anne, she leaves after receiving a kiss on her cheek from Peter. The kiss definitely symbolizes growth and intention towards their new relationship as they leave their more immature one in the past.