The post-apocalyptic narrative of The Road portrays a bleak world, indeed, in which the man and the boy struggle to stay alive only because of an instinctive drive to do so.
This paragraph in the first person seems to be written in the stream-of-consciousness format. The man recalls that he was going to catch the dog for food, but his son cried, entreating his father to not kill it.
That is the dog he remembers. He doesn't remember any little boys.
The boy remembers the dog because the father saw it as only food, not the "friend of man." The dog, a dumb creature innocent of everything was about to be made a victim and the boy cannot bear it.
On page 67 of the Vintage edition of The Road, there is the description of the confrontation of the father with another man who has a truck. When this other man grabbed the boy, the father shot him and the flesh of that man splattered upon the boy's forehead.
The man fell back instantly with blood bubbling from the hole in his forehead. The boy was lying in his lap with no expression on his face at all....He wiped the blood from his face and held him. It's okay, he said. It's okay.
Perhaps, then, the boy recalls this incident in his plea to not kill the dog, and in his mind, the father reflects upon their state of existence.