The "Negro problem" is characterized by Du Bois as a "veil" that separates black and white consciousness. Black citizens necessarily develop a "double consciousness" in US society in which their own self-perceptions are at war with the way they are perceived by white society.
African Americans therefore become a problem to white citizens because these white citizens refuse to fully accept them as equal human beings, instead marginalizing them as "others." This makes it difficult for black citizens to integrate into American life and a struggle for them to develop a positive self image and strong sense of self. African Americans, Du Bois contends, live double lives, presenting one image—most often the expected image—of who they are to white citizens while living and feeling a very different way.
At the beginning of The Souls of Black Folk, Du Bois mentions being treated as a "problem" for being black. He says that in response to the implied question whites often dance around, he seldom answers:
To the real question, How does it feel to be a problem? I answer seldom a word.
A little later in the introduction, however, he does try to answer the question, by saying it is painful to be considered a problem. This othering makes it difficult for African Americans to find a true sense of identity and worth. He writes:
the Negro is a sort of seventh son, born with a veil, and gifted with second-sight in this American world,—a world which yields him no true self-consciousness, but only lets him see himself through the revelation of the other world. It is a peculiar sensation, this double-consciousness, this sense of always looking at one’s self through the eyes of others, of measuring one’s soul by the tape of a world that looks on in amused contempt and pity.
Du Bois spends much of his book arguing that the way forward for black citizens is not to accommodate themselves to second class citizenship but to fight for full civil rights and equality with white citizens.
The "Negro problem" is not truly a problem for black people but for whites who struggled with two things in the aftermath of the Civil War—how to grant full citizenship to a people whom they had kept in permanent bondage for centuries, and how to acknowledge these people as the visual reminders of the nation's sin of slavery. DuBois mentions how guilt about the latter prevented white people from properly dealing with the former issue. Thus, they never ask him about the "problem" directly:
BETWEEN me and the other world there is ever an unasked question: unasked by some through feelings of delicacy; by others through the difficulty of rightly framing it. All, nevertheless, flutter round it. They approach me in a half-hesitant sort of way, eye me curiously or compassionately, and then, instead of saying directly, How does it feel to be a problem? they say, I know an excellent colored man in my town; or,...
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I fought at Mechanicsville; or, Do not these Southern outrages make your blood boil? At these I smile, or am interested, or reduce the boiling to a simmer, as the occasion may require. To the real question, How does it feel to be a problem? I answer seldom a word.
Notice how DuBois frames himself as a calm observer of the distress that whites seem to have around the subject of black citizenship. DuBois, arguably, answers "seldom a word" because he knows that the people he encounters do not really want to tackle the issue. What they would prefer, instead, is to present themselves as hospitable to the black presence ("I know an excellent colored man in my town") or to behave as though racism is a distant and peculiar Southern problem ("Do not these Southern outrages make your blood boil?").
The "Negro problem," as Du Bois describes it, is rooted in his desire to "make it possible for a man to be both a Negro and an American." He describes a sort of "double consciousness" experienced by African-Americans. Black people born in the US were not Africans—their roots in the United States dated back generations. Yet they were not fully Americans, because they were unable to exercise many of the most basic rights enjoyed by white Americans, and their cultural existence was not valued or considered part of an "American" culture. Addressing the "Negro problem" was, Du Bois said, "a concrete test of the underlying principles of the great republic." It is true that Du Bois says he declines to answer the question, posed at the beginning of the book, of how it feels to be a problem, and no doubt Du Bois did believe that this question tended to deny his humanity. Yet the question is an important one to Du Bois, for The Souls of Black Folk is, in large part, meant to answer it. He urged his readers to "listen to the striving in the souls of black folk," and the "striving" in question was toward an answer to this question. In the second chapter of the book, he makes it clear that the burden of addressing this problem was not limited to African-American men and women: "The problem of the twentieth century," he said, "is the problem of the color-line."
The negro problem, as it was called, was the issue of what the place of African Americans in society should be. They were no longer slaves, but the vast majority of white people did not consider them to be equal. So what place was there for them between slaves and equals? As James Baldwin once said:
At the root of the American Negro problem is the necessity of the American white man to find a way of living with the Negro...
I'm assuming that you're talking about the very beginning of the book where he says "How does it feel to be a problem. I answer seldom a word."
I believe he says this because he resents being seen as a problem and not as a person. He resents the way that people try to pretend that they accept black people ("I know an excellent colored man in town.") even when they really see him as a problem.
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