There are a few layers to DuBois's famous quote. On one level, DuBois is discussing the idea that the trajectory of post-Reconstruction America was one which was highly undemocratic in nature. Although white Republicans and Democrats would claim to be living in a democracy, the US under Jim Crow was a polity that systematically disenfranchised a sizable minority of the population based on race. DuBois's formulation holds that this fundamentally undemocratic and at odds with any notion of representative government or freedom.

The second layer of DuBois's analysis was that the institution of a neo-slavery in post-Reconstruction America (eg chain-gangs, share-cropping, debt peonage) was part of a larger process of a racialized system of global economic exploitation. DuBois notes:

The whole weight of America was thrown to color caste. The colored world went down before England, France, Germany, Russia, Italy, and America. A new slavery arose. The upward moving of white labor was betrayed into wars for profit based on color caste.

What DuBois is doing here is citing the then-quite recent First World War and how it was a war caused by imperial and economic competition. This was a theory popular in the interwar period, and now a quite a bit discredited (although there are partisans out there as seen in this exchange in the NYRB) and DuBois contends that while white workers have temporally benefited from Jim Crow and imperialism, they had to pay a high price when conscripted to fight in a war that they had no real stake in winning. The Depression, which was ongoing when Black Reconstruction in America was published, further underscored the idea that white labor accrued no tangible benefits from Jim Crow and other racial caste systems. Whites might feel superior to other races, but their bellies would still be unfilled during an economic downturn.

This is the underlying political goal of DuBois's scholarship. When he speaks of the "hearts of black folks," he is invoking an idea that the way out the mess America and the rest of the world had found itself in was to abandon these racial caste systems. This according to DuBois would be powerful especially since the white laborer had much more in common with his black counterpart than his white bosses. Black Reconstruction in America traced out a moment in American history- Reconstruction- where black and white labor in the South tantalizingly came close together to form a radically different and more democratic society, but fell apart as violence and racism formed a wedge between these groups. DuBois held that while white society had forgotten this moment, black society had not.