In Fahrenheit 451, when Montag says, "they just might stop us from making the same damn insane mistakes," what is he talking about?
The quote you have identified comes from a conversation that Montag has with Mildred after he has been awakened to his own sense of dissatisfaction with his life and with the world, and has begun to think that books are perhaps one solution to his issues and the issues that his world faces. He tlaks just before the quote of the idea that books are what can "get us half out of the cave" of ignorance that has been created, and your quote refers to the way that books chronicle the history of humanity and the kind of mistakes and lessons that humanity has learnt.
The quote therefore refers to books and the knowledge they contain that is essential for humanity. Montag is of course completely right. If you remove books and history, there is no record of humanity's struggles and challenges throughout the ages, which results in the lack of knowledge and ability to face the issues of humanity today. This is the problem of Montag's society, and his awareness of this makes him work as a rebel against the system.