The men Montag meets at the end of the story are "living libraries." How are they preserving the knowledge that their civilization has burned? Why is it important for them to do so? How is that knowledge important to the future?

These living libraries preserve knowledge by studying memory books.

Explanation:

 Your question refers to the book Fahrenheit 541 by Ray Bradbury.

These men who are living libraries preserve the knowledge they believe is necessary for the future. It is important that they do this because if they don't, that knowledge will die forever.

They plan to pass the books on to their children, knowing that much of that information will be lost but still very useful.

Each man had a book that he wanted to remember and it was thus that over the years they were setting up an organization.

Let's look at the following quote:

"And when the war's over, some day, some year, the books can be written again, the people will be called in, one by one, to recite what they know and we'll set it up in type until another Dark Age, when we might have to do the whole thing over again. "

faithful servants eumaeus and philaetius:

his ragged vest then drawn aside disclosed the sign conspicuous, and the scar exposed: eager they view'd, with joy they stood amazed with tearful eyes o'er all their master gazed:

these servants know ulysses by the scar he got when he was a child - so when he showed them the mark, they immediately knew this is truly their master and they hugged him, finally starting to believe his words.