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.