The Southern college that the narrator attends is modeled on Tuskegee Institute in Alabama, which was founded by Booker T. Washington. This detail is important because, like Washington, Bledsoe relies on wealthy, white philanthropists (Mr. Norton), to fund the college.
Bledsoe's intent in hiring the narrator to take Mr. Norton around was to show Norton all of the good that his money could do for "uplifting" the race. When the narrator digressed from Bledsoe's instructions and took Mr. Norton out to the old slave quarters where they met Trueblood, he showed Mr. Norton aspects of black identity—of human identity, really—that Dr. Bledsoe did not want the white man to see.
Bledsoe's problem with the narrator is that the young man did not have the "sense to lie." Though he insists that Norton wanted to go out to the old quarters, Bledsoe counters that black people have to know how give white people what they want by lying to them:
Why, the dumbest black bastard in the cotton patch knows that the only way to please a white man is to tell him a lie! What kind of education are you getting around here?
In this respect, the narrator had also failed, at this point, to learn the lesson of his grandfather's deathbed speech—to "undermine 'em with yeses." When the narrator threatens to fight the college president's threat of expulsion, Bledsoe is calm and amused:
Tell anyone you like. I don't care. . . . I's big and black and I say 'Yes, suh' as loudly as any burrhead when it's convenient, but I'm still the king down here. . . . The only ones [white people] I pretend to please are big white folk, and even those I control more than they control me.
He goes on to explain that even if Norton says that he does not want the narrator punished, he really does, and he will when Dr. Bledsoe, who believes that he has insight into the true desires of white people's interests, tells him that the narrator should be punished. According to Bledsoe, the notion that the narrator is lying will appeal to Norton's desire of what he wants to be true. What Bledsoe is implying is that Norton wants to believe that black people are dishonest and out to do harm. If that lie helps to keep Bledsoe in power, he will tell it: "I'll have every Negro hanging on tree limbs by morning if it means staying where I am."
The answer to your question goes beyond the previous educator's mention of internalized racism (the feeling of being inferior) and oppression among black people. Bledsoe is a man who has learned how to manipulate white racism and supremacy to his advantage, not unlike the house slave who improved his conditions by telling on disobedient slaves or even lying about their actions to please the master and reinforce his negative ideas about black people. Bledsoe has constructed an image of himself as exceptional—as one of the "good ones"—an image that relies on black people, collectively, being bad or less worthy.
Bledsoe expels the narrator to maintain his power, which is clearly not as self-assured as he thinks it is, otherwise he would not have to tell lies about black people to reinforce white supremacy. He expels the narrator for being poorly conditioned and not understanding that any power that a black person has in the South—in the entire country, really, at this time—relies on knowing that you cannot really act as an individual, but only in the service of white interests.