Normally, whenever the father and the son encounter a survivor of the rough world that they are living in, the father discourages offering any help whatsoever. It might seem cruel, but it is really a survival tactic. First of all, they only have a limited amont of food, and to share any food brings death even closer. Secondly, the more people they have in a group, the harder it is to hide and remain inconspicuous, which is necessary to survive against the barbarians. And lastly, the father doesn't know if the survivors that they encounter are barbarians or not; they've been known to lay traps, and he's afraid if they help, it will just bring out barbarians who were lying in wait, watching. So, the father wisely discourages any form of help. However, that doesn't stop the son from wanting to help. He still has goodness, faith in human nature, and hope inside of him, and his heart reaches out to others who are suffering. So, it's a battle between the father and the son every time.
The father agrees to help the old man they encounter for several reasons. One is is that the old man just looks so pathetic, weak and decrepit. It was hard not to feel for him--being so old, it would be so hard to survive. Secondly, the son managed to convince him; he let the son win one battle. Another reason is that they followed him for quite some time, to assure themselves that he wasn't a planted trap, and was quite alone. Then, the old man himself is terrified of the father and son; this indicates that he himself is not a barbarian, but afraid of them, just like him. He is weak, old, almost rotting as he stands, and the father has mercy on him.
Even though the father does agree to help him, he still puts limits on their help; he doesn't let the boy touch him, or give him a lot of their food. He doesn't let the boy "keep him," as he asks; they just stay with him one night and move on, feeding him what they can. It's a rough world that they live in, and the father and the old man are wary and suspicious, but come together briefly, for a moment, before they move on in their own struggles. I hope that helped; good luck!