The Diary of Anne Frank, also known as The Diary of a Young Girl, was written by a young Jewish girl who was hiding with her family from Nazi persecution. She kept the diary from June 1942 until August 1944, when she and her family were arrested and sent to concentration camps.

Anne's comment that "paper has more patience than people" is in the entry for Saturday, June 20, 1942. This is only a few days after she began writing in a red and white checkered blank book that her parents had given her for her thirteenth birthday.

The quotation is part of the explanation she gives for starting and keeping a diary. She says that though later nobody may be interested in the writings of a thirteen-year-old girl, she has some thoughts she feels compelled to express.

I feel like writing, and I have an even greater need to get all kinds of things off my chest.

She confesses that she is surrounded by all sorts of loving people such as her family and other acquaintances, but she doesn't have a true friend in which she can confide.

All I think about when I'm with friends is having a good time. I can't bring myself to talk about anything but ordinary everyday things. We don't seem to be able to get any closer, and that's the problem.

For this reason, Anne invents an imaginary friend named Kitty and addresses the diary to her. The paper in her diary book, represented by Kitty, unlike the real people she knows, has endless patience to listen to and record anything she wants to say. That's why she says that "paper has more patience than people."

Anne’s point is that people can sometimes be impatient with her, but she can write whatever she wants in her diary and complain all she wants to, making her diary seem understanding and compassionate.  Anne describes a saying that “paper has more patience” when she considers that she has never written in a diary, but that writing in a diary allows her to write her innermost thoughts.

Yes, paper does have more patience, and since I'm not planning to let anyone else read this stiff-backed notebook grandly referred to as a "diary," unless I should ever find a real friend, it probably won't make a bit of difference. (Saturday, June 20, 1942) 

Since Anne and her family are in hiding from the Nazis, there is often a lot of conflict. Too many people are in close quarters there.  Even before hiding, Anne and her mother often did not see eye to eye.  Anne felt that her mother was too critical, and her mother felt that Anne was immature or rude.  

The diary was an outlet for Anne.  She could write in it when something upset her, and write about her hopes and dreams.  That is why she named it Kitty, and said it was her only true friend.