I consulted the Dutch edition Het achterhuis: dagboekbrieven 12 juni 1942 - 1 augustus 1944 (Edited by Anne's father Otto Frank and Mirjam Pressler. Amsterdam: Prometheus, 2010). This edition of the diary does not contain an entry for 7 November 1942; there is nothing between the entries of 5 and 9 November 1942.
However, there are four (!) versions of Anne Frank's diary:
- Version A is Anne's original diary, which she started writing on 12 June 1942.
- Version B is a reworked version by Anne. She started working on this in February 1942, after a Dutch minister in exile (in London) said on the radio, that after the War, testimonies about the suffering of the Dutch should be collected and published. Version B was a version destined for publication; while working on it, Anne continued writing version A.
- Version C is a version that Anne's father Otto Frank (the family's only war survivor) published in 1947. This was a shortened version based on versions A and B.
- Version D is the version that is usually published (and translated) today.
The diary entry for 7 November 1942 does not exist in version A; it is something that Anne added in version B. This is a long letter about how Anne's parents habitually side with her older sister Margot when there is some sort of conflict. This leads her to the question whether God is testing her and whether she should try to become "good" without guiding examples. This then leads to the paragraph cited in the question.
The diary entries are written as letters to "Kitty"; in an early diary entry from 20 June 1942 (actually, the first entry in version B) that she did not have a true friend, i.e. someone with whom she could talk about intimate things, so her diary might serve that role. So the diary entries are not simply observations with a date above them but letters to the diary/Kitty with a salutation and a closing line. So the claim, in Yaskskier's answer, that "letters" in the quote refers to "letters of the alphabet" is factually wrong. (The Dutch word for "letter" in the sense of written message is "brief"; the Dutch word for "letter (of the alphabet)" is "letter", so there can be no mistake.)
The statement that the letters will not be read by anyone refers to the fact that she would never let anyone read her diary (or "diary letters", as the title of the Dutch edition says). This statement comes from the 20 June 1942 entry (the first) in version B and therefore refers to version A.
The comfort she does not get refers to her relationship with other family members. The 7 November entry also says that Anne is in many respects the opposite of her mother and asks whether anyone could really satisfy their children.
She writes that she wants to improve herself in order to satisfy the demands by other people. She probably also hopes that this would help her overcome the unequal treatment of her and Margot.