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.
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."