The idea is that the constant stress imparted to black people by institutional and systemic racism leads to poorer health outcomes.
This doesn't disprove that white people don't also have worse health outcomes with increased stress, but the focus for many researchers is in targeting health disparities and social inequalities. It just so happens that race plays a big role in that.
Success is harder for black folks. If they don't leave the poorer areas they grew up in, did they really succeed? There's an increase risk of being a victim if you are richer than most in a poorer area. if they leave, they are socially isolated from their comfort zone and friends, labeled a sell out, have to conform to social norms of the new environment. In both cases friends and relatives constantly beg for money. The worry of losing it all is very real. Yes, life is harder for blacks in America. A black African that succeedes in Africa doesn't have as much stress.
You'd probably also be interested in "The short and Tragic Life of Robert Peace", a young man who went from inner city poverty, to a degree in molecular biophysics and biochemistry from Yale university, to being killed in a drug robbery.
I assumed, on reading, that they were suggesting it was due to some societal factors (e.g. racism) that makes it harder for black people, but maybe I misinterpreted it.
Well, even as a poor white, fighting poverty etc, it's different than dodging bullets in your neighborhood (as was often the case, in the time of the study), plus, the dominant race of society seeing you as criminal/inferior/threatening.
If I understood the article correctly, the effect was originally associated with African Americans, but has recently been extended to others in similar "striving" conditions.