The Diary of Anne Frank is as famous as it is for literary and non-literary reasons.
The literary reasons are its honesty and the quality of its descriptions of character and emotion.
To be frank, those qualities are probably less important than its non-literary qualities. It was published soon after World War II, in a time when the entire world was still trying to make sense of the Holocaust. It was therefore the first firsthand account many people read of life during the Nazi regime. In many cases, it is the only one read.
Frank's character is easy to identify with; she seems likable and innocent. To read about such a person staying as positive in the midst of what we know to be horrific makes the account, and her, deeply sympathetic.