What is the negro problem? Why does Du Bois say he can't or won't answer when he is asked about the problem?

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.