How does what might be called the legacy of the Civil War as it operates in the South operate in “Barn Burning?”
The legacy of the civil war might be quite different depending on who you ask. I assume that is why you specifically mentioned "The South." In the North, and in most history books, a legacy of the Civil War is that it preserved the Union of the United States. There are other positive legacies as well. An end to slavery, which is good, but by no means were race relations positive after the war.
For the South though, the initial legacy wasn't so brimming with positive reflections. Bitterness was most definitely there. The south lost and losing doesn't feel good. What's worse about that loss is most Civil War historians would agree that the South had a better military. Soldiers were better with weapons and horse riding. Better commanding officers too. The south won a lot of battles. An advantage the south didn't have, though, was population numbers and the manufacturing and transportation capabilities of the north. The south probably felt that the north got a "cheap win."
On top of general losing bitterness, the entire infrastructure of the south was destroyed. Sherman wiped out just about everything in his path, and the rest of the Union army wasn't much different. And the south's entire labor force had been freed. To the south, they lost, lost their way of life, and lost their ability to make a living. They got poorer while the north was just getting richer.
In "Barn Burning" that bitterness toward the wealthy is exemplified in the character of Abner Snopes. He's been broken, literally, by the war. His leg is lame. He's been forced to do the job that slaves once did, and he's making hardly any money for it. He sees the men that he works for as wealthy land owners that are just getting richer off the sweat of his back while he is scraping to get by. He can't do anything about it other than lash out with violence and aggression (which is the only way he knows how to fight back). So when he feels that he has been wronged, he burns barns down or drags poop onto rugs, or ruins them in the cleaning process. He's bitter toward his oppressor in the same way that the south feels bitter toward a loss to the North.