This quote from The Diary of Anne Frank implies that she finds it easier to be honest in writing (in her diary) than to speak openly with other people. It is true that a journal asks no questions, does not interrupt us, or offer feedback. A journal or diary is effectively a perfect listener and confidante. Anne struggled with a lot of feelings she was afraid to tell her family for one reason or another, so putting these thoughts to paper offered her some relief from them. Sometimes it is the act of writing or saying something to release it that is more valuable than the words actually being heard.
Anne was sometimes criticized by her family for being too talkative, so it must have been nice for her to be able to express her thoughts and feelings in a way that was not subject to criticism. In the same passage where she shares the saying, "Paper is more patient than man," she admits to feeling like she doesn't have any real friends she can be honest with. While she loves her family and the friends she does have, she feels that these relationships are superficial and that she cannot be truly honest about her feelings.