What is significant about the statue at the end of this story?

The statue of the "artificial" black man brings Mr. Head and Nelson back together again at the end of the story. It is a symbol of God's mercy and grace.
Before seeing the statue, Mr. Head and Nelson, two country people, have gotten lost in Atlanta. Because the boy earlier complained that Mr. Head didn't know his way, Mr. Head decides to teach him a lesson. He hides around a corner in an alley while Nelson naps, and when Nelson wakes up, he is so frightened to be alone in the city that he runs wildly, knocking over an old woman and scattering her groceries. When Mr. Head comes forward, women begin to accuse him, saying that he will have to pay for the groceries, that the old woman's ankle is broken, and that they will get a police officer.
Mr. Head is so terrified that he denies knowing Nelson, just as Peter denies Christ:

"This is not my boy," he said. "l never seen him before."

Nelson and Mr. Head are allowed to go, and no policeman gets involved, but Nelson is so upset and angry that Mr. Head pretended not to know him in his time of greatest need that he rejects Mr. Head coldly. Nelson is angry and unforgiving. Mr. Head, who has no one else, is overwhelmed with pain at the idea of forever losing Nelson's love and respect:

He felt he knew now what time would be like without seasons and what heat would be like without light and what man would be like without salvation.

However, when both Nelson and Mr. Head stop to look at the statue of the black man, they are reunited over it. They both seem to see something of their own brokenness in its damaged form:

They could both feel it dissolving their differences like an action of mercy. Mr. Head had never known before what mercy felt like because he had been too good to deserve any, but he felt he knew now.

Mr. Head feels all of God's mercy, forgiveness, and salvation as the statue heals the rupture between them.
It is typical of O'Connor to choose something as tacky and grotesque as a worn statue to bring God's grace into a character's life. It functions like a madonna or the statue of a saint. O'Connor is showing that God works in mysterious ways and that his grace is everywhere.