In this book, as in his entire life, Washington felt that the most important thing for black people was to be educated and hard-working. The education that he felt was important was mostly vocational education that would allow blacks to be good workers. From this, we can infer that Washington felt that the cause of the race problem was the fact that blacks were insufficiently trained and insufficiently industrious.
To Washington, the only real solution to the race problem was hard work. He wanted blacks to "cast down their buckets" where they were. He wanted them to accept that their place (at that point in history) was to do the hard work of agriculture and mining and domestic service. He felt that blacks who were properly educated and properly hard-working would win the respect of whites.
Washington does not explicitly say what he thinks the cause of the race problem is. But we can infer it from what he encourages blacks to do. He clearly feels that the cause of the problem is that whites do not respect blacks and that this lack of respect comes from blacks' lack of education and industriousness.