The "lesson" that Jefferson demonstrates is that human beings have the capacity to change. Jefferson's evolution underscores the novel's central theme of transformation.
Jefferson's execution is justified because he is not seen as a human being. The refrain of "he's a hog" is used to dehumanize him, making his death easier to accept. The sheriff underscores this philosophy when he tells Grant that he would "rather see a contented hog go to that chair than an aggravated hog." The sheriff believes that "there ain't a thing you can put in that skull that ain't there already." The world that sentences Jefferson to die believes that individuals like him are incapable of change.
Jefferson changes because of Grant's instruction. The lessons that Jefferson gains are ones that increase his awareness of the world and his place in it. Prior to his instruction, Jefferson believed his identity was static. He absorbed the mentality of the society around him. This can be seen in his criminal life. Jefferson was not an active participant in the robbery. He did not plan it. Rather, he went along with it, losing a bit of his humanity because he essentially believed he was incapable of change. The human ability to grow and "be a man" is something that Grant underscores in his lessons: "And that's all we are, Jefferson, all of us on this earth, a piece of drifting wood, until we—each one of us, individually—decide to become something else."
As a result, Jefferson learns that he can change. He learns of his own humanity. Jefferson shows this when he writes that "Man walk on two foots; hogs on four hoofs." Lessons like these enable Grant to passionately articulate his human condition: "tell them im strong tell them im a man good by mr wigin." In asking Paul to deliver this message, Jefferson has embodied change, confirmed when Paul says that "Jefferson was the strongest man in that crowded room." The lesson that both Jefferson and Grant learn is the capacity of human growth.