What do the man and boy mean by "Carrying the fire"?
This a great book and a good place to start is by looking at the context. Here is full quotation:
“You have to carry the fire."
I don't know how to."
Yes, you do."
Is the fire real? The fire?"
Yes it is."
Where is it? I don't know where it is."
Yes you do. It's inside you. It always was there. I can see it.”
The point here is at least threefold. First, it is a learned reference to Prometheus, the Greek god who gave fire to people. This enabled people to create civilization. This reference, then, is a very appropriate, because what is needed is a new civilization.
Second, in this post-apocalyptic world, fire represents what is good in view of the darkness. Fire, therefore, is symbolic of what is noble and what it means to be human. This is important, because in this setting people are forgetting what is mean to be human. It is a world where people are only for themselves.
Third, these are the words uttered to the boy at the father's death. To keep the fire going means to press on. Don't give up. He wants his son to keep up the fight to create a new humanity.