The answer to this question relates to the pervasive theme of grace in this story and the way that grace is shown to offer both the grandmother and the Misfit a second chance. Let us remember the kind of person that the grandmother is. She is a profoundly selfish individual who completely lacks any self-awareness about her own mistakes and faults. She deliberately manipulates her son's family so she can get her own way and is not above lying and deceiving either, as is shown when she smuggles in her cat against her son's explicit instructions. However, as she talks to the Misfit, and in particular as they discuss the character of Jesus and his actions, she experiences something of an epiphany as she reaches out to the Misfit. This of course prompts her to say: "Why you're one of my own babies. You're one of my own children!" This is rather a confusing statement, as quite clearly the Misfit is not her child. But let us remember that these words are actually a realisation that both she and the Misfit are human beings who are subject to mistakes and sins. This similarity that she sees between herself and the Misfit could actually said to be the one moment in the story when she sees herself for who she is.
Of course, this change in the grandmother and the moment of insight she is given before dying is paralleled by the Misfit's own change. Before, he declared that there was "no pleasure but meanness" in life, but at the end of the story, he stops Bobby Lee from rejoicing in the "fun" they have had by saying "It's no real pleasure in life." Grace is shown to be able to descend on the unworthiest of recipients.