In Flannery O'Connor's "A Good Man Is Hard to Find," why does the grandmother have an epiphany, and what is its significance?

In Flannery O’Connor’s short story “A Good Man Is Hard to
Find,” the grandmother’s epiphany – or moment of enlightenment or revelation – is almost
surely the moment when she reaches out and touches The Misfit. Although The Misfit and
his henchmen have slaughtered all the other members of her family, the grandmother is
nevertheless able to suddenly see a connection between herself and The Misfit.  She thus
reaches out and touches him while saying,

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“Why, you’re one of my babies. You’re one of my
own children!”

As soon as she
says this, however, The Misfit shoots and kills her.

This
startling moment in the story is also one of the story’s richest moments for a number of
reasons, including the following:

• The
  grandmother had earlier been estranged from her biological son, Bailey, and his wife and
  children. The fact that she is able to see and make some connection with The Misfit is
  therefore a sign of a significant change in her entire
  existence.


• The grandmother, early in the story, had been
  unable to see any real connection between herself and a small black child the family had
  passed along the side of the road on their trip to Florida. Now, however, she is able to
  see some real connection between herself and The
  Misfit.


• Throughout the story, the grandmother had assumed
  that she was quite different from people such as The Misfit. For example, when
  discussing contemporary problems with Red Sammy Butts, she had actually said that the
  entire continent of Europe was to blame for any problems that existed in the world [!]. 
  This is one of many unintentionally funny statements she makes during the course of the
  story. The statement implies her tendency to consider herself a good woman who has no
  responsibility for any problems, either within her own family or in the world at
  large.


• The fact that the grandmother is finally able to
  see some connection between herself and The Misfit suggests that she is finally also
  able to see some connection between herself and evil, which is one thing The Misfit
  surely symbolizes.


• However, the fact that the grandmother
  is finally able to see some connection between herself and The Misfit also suggests that
  she is finally able to really put into practice the Christianity she prattles about
  elsewhere in the story -- a Christianity she never really lives, in
  the truest sense, until right before she dies.  She reaches out to The Misfit in love
  and compassion and fellowship. The fact that he kills her for doing so means little to
  O’Connor.  What matters is that the grandmother has actually, for once, acted like a
  true “grand mother.” The grandmother’s physical death is insignificant: we will all die,
  but few of us will ever live a moment as authentically rich and full of meaning as the
  grandmother’s last moment on earth. In the last split seconds of her life, she finally
  is used, by God, to give The Misfit a much-needed epiphany of his own. How he chooses to
  respond to that epiphany is, as O'Connor herself once said, another
  story.