How would you account for the actions of the father, who burned the first barn, went through the rug incident, and is going to burn the second barn? What is motivating the father to do these things?
Abner is a scary guy. He rules his family with an iron fist both physically and psychologically. At one point in the story, Abner takes his son Sarty to the side. Abner tells Sarty that he should never betray his family and blood, and then hits Sarty because he knew Sarty was going to tell the truth in court after the first barn burning.
Abner is a broken man. Perhaps that is why he is so angry all the time. First, he is broken physically. He has a lame leg that he drags behind him. Abner is also a broken man psychologically. He feels beaten down by the class system that he is a part of. He is a sharecropper who is forced to move from plantation to plantation. He works the land and his only real payment is a small portion of the crops that he raises. He and his family are barely scraping by in this system. Because of his body, his lack of wealth, lack of options, and no real positive future, Abner has become a bitter, angry, violent, and abusive man.
The reader doesn't really have any specific insight into why Abner burned the first barn down, but after reading the rest of the story, it's clear that his reasons for the first barn were probably not much different from his reasons for the rug and second barn.
He sees the big huge beautiful white house and its owner as a symbol of everything that is beating him down. De Spain has made his wealth on the sweaty backs of those under him. He gets richer and Abner gets poorer. The rug incident is Abner's way of telling de Spain what he thinks of de Spain and his wealth. The second barn is Abner seeking vengeance against de Spain for his 10 bushels charge for the damaged rug.