The "full" argument goes:
We as a government are actively trying to make animals grow up and become independent of people by not giving them food (handouts). Food stamps (handouts) are the same thing, and by giving them to people they are not learning to grow up and become independent.
It stops there for most GOP, but then there is the conspiracy theory that it's to enslave the human population, etc. It just kind of gets ridiculous from there.
I think it's mostly a pandering to their base. Their base is just a bunch of old people who don't understand that the government has checks in place to prevent that sort of thing, and also (probably, I don't know about food stamps specifically but other programs work this way) wean you off slowly to make sure that you can get independent if you can.
These people also don't understand disability at all, and think that all mental illness is just something you can snap out of. They relate their "sad" moments of life (losing their dog, their grandparents, etc), to being mentally depressed, and realize that they could get over these events so everyone else should, too, right?
Most GOP that go down this trail assume that the entire human race and biology works identically for each human being, because we are the same species. Except black people and muslims, who are not quite the same species.
I want to make a disclaimer, none of this is my opinion. This is just what I learned growing up in a highly conservative family.