What does Anne mean when she says paper is more patient than man?

What Anne means by this is that she can be more honest in writing her diary than she can in communicating with other people. She can put down on paper her innermost thoughts, hopes, fears, and dreams, sharing things with her diary that she could never share with anyone else.
This is crucial for someone of Anne's age, especially given her situation. As a teenage girl, she's going through a difficult phase in life, with all the usual problems of adolescence. To make matters even more complicated, she's forced to spend her early teenage years stuck in a cramped annex in order to avoid detection by the Nazis. And while all this is going on, Anne, like so many other girls of her age, feels unable to confide in her parents. That's where the diary comes in. This little book has infinite reserves of patience. It will always be there for her, always ready for whatever Anne has to say, and that's not something that could be said of anyone in her life.