The grandmother's actions at the end of the story are quite interesting; she spends most of the story, and we can easily imagine, her life, being a nasty, manipulative, condescending woman stuck in old-fashioned ways and expectations. However, at the end, she turns loving and kind. Perhaps, faced with her own mortality, she is finally able to realize that all people essentially, are connected, and she reaches out to share that near-death revelation. Flannery O'Connor often told stories that had a theme of religious redemption, even though the stories themselves were pretty aggressive and often brutal in their violence. Most critics agree that at the end, as the grandmother faces the end of her life, and realizes that she is going to die, that she has a sort of "vision," or clarity. Before the declaration that she stated, she refers to Jesus, and how Jesus loves everyone; perhaps this is an attempt to ease his anger, to manipulate him into not killing her, a desparate attempt to save her life. Or, maybe she herself is realizing that she had been a pretty prejudiced and mean person her whole life. But, as the Misfit argues with her about Jesus, getting more angry, he gets in her face. It is at this point that she seems grasped by a force that is very much unlike her own, which causes her to reach out in love and declare "You're one of my babies. You're one of my own children!" She is probably realizing that he was, like she herself was, a child of God. She could have been referring to herself being the mother, in a figurative sense, of the Misfit, and feeling maternal love for the man, or, she could be speaking for God and religion in general, proclaiming that he was God's child. Either way, she realizes, too late probably, that she is connected with this man, and as she dies, is filled with love for him, instead of the bitter, cynical thoughts that characterized her life to that point.
I hope that those thoughts helped; good luck!