This question can best be answered by referring to the change in the character of Grant as the novel progresses. He begins the novel deeply unhappy about the racial segregation that is seemingly such an intrinsic part of American society. This is compounded by the way that the defence attorney, trying to save Jefferson's life, dehumanises him and compares him to a hog:
What justice would there be to take this life? Justice, gentlemen? Why, I would just as soon put a hog in the electric chair as this.
The tragedy and injustice of this comparison impacts all of the characters, but it is interesting that Grant, in spite of his indignation, is very cynical about the potential he has to change society. He dreams of leaving his job and moving away and taking the easy option of escaping.
However, through his relationship with Jefferson that he cultivates thanks to the pressure of his mother, Grant comes to realise that adopting his cynical approach to life and society is the same as giving up and dying, and that even the most tiny triumphs can result in transformation. He changes his view of Jefferson from being a hopeless cause and begins to see him as a fellow human being whose plight Grant takes on as his own personal battle. The story's title is of course overtly referring to the "lesson" that Grant gives Jefferson, but as much as Jefferson learns about the need to be strong as he faces his execution from Grant, the real "lesson" is the lesson learnt by Grant as he realises the importance and the power of civic responsibility. What it is to be American in this novel is therefore shown through the character of Grant and his gradual integration into the society that he starts of by despising so greatly.