Dealing with the grandfather's deathbed speech to his son wherein he tells his son to live "with your head in the lion's mouth" and that "our life is a war and I have been a traitor all my born days," we can point again to an important contrast between the grandfather's highly developed self-awareness and cynicism and the narrator's blind sense of faith in the good intentions of others.
The grandfather suggests with his literal last words (as opposed to his final words in the chapter, dealt with above) two essential ideas. First, Blacks and Whites exist on two sides of a conflicted divide. Second, the best course of action and attitude for a Black man to take is to be always aware of this divide.
The narrator does not understand such a point of view but the course of his life as told in the novel demonstrates what his lack of understanding can lead to as he is exploited and betrayed by those who he believes to be on his side.
The narrator of the story in Ellison's Invisible Man is very different from his grandfather. Where the narrator is naive and hopeful that he will be able to succeed and get ahead on the terms set by those in power, the grandfather is deeply cynical. He does not trust White authority and does not, it would seem, believe in any real altruism or generosity on the part of that same authority.
When the narrator is subjected to electric shock, humiliation and physical punishment before giving a speech praising an race politic of appeasement and humility, he is entirely unaware of the irony of his situation. He goes home with a scholarship and a briefcase and is happy.
The narrator has been co-opted, to use a term from critical theory. He has identified his own goals with those selected for him and preferred for he and his kind by the White authority. He has, basically, agreed to be patronized by interpreting the events of the evening in a compartmental fashion.
Far from regretting the terrors of the evening, the narrator reflects only on his "triumph" at receiving a scholarship.
"I was overjoyed; I did not even mind when I discovered that the gold pieces I had scrambled for were brass pocket tokens advertising a certain make of automobile."
As long as the narrator is willing to accept the gifts and praise of White men like these, he will be subjected to their terrors as well (just as he was in the course of the evening).
This is what the grandfather's last words are intended to convey. The narrator will be kept running in ways that parallel the painful and humiliating ways he was kept running in order to earn his briefcase.