Up until about 1971 (possibly later, but that's the date of Loving vs Virginia), the US had formal, legal discrimination against black people. On finally removing that, there was at various times discussion of whether people who had been legally discriminated against all their lives should be compensated, as if they had been wronged in a tort sense. This would obviously be extremely expensive, and anyway impossible to quantify, so it never happened.
Harvard, and many other colleges, has a big base of "legacy" admissions, as well as a certain amount of generational knowledge and connections - you're more likely to get into Harvard if your parents went to Harvard, over and above mere class status. Since black people were under-represented in this category, people came up with the idea of putting a thumb on the other side of the scale and artificially increasing their rate of admission. Can it balance exactly against the disadvantages of discrimination? No.
> I genuinely don't get why the race thing is part of the equation. Shouldn't this just be run-of-the-mill social democratic "lets hand out some extra opportunities/benefits to the poor" program?
America is not in the least social-democratic, but racism and anti-racism have been there since the beginning and will probably dominate US politics until the last person who remembers the KKK is dead.
(It could be worse: Haitians who had freed themselves from slavery by the mass murder of their oppressors were made to pay a huge amount of compensation to them!)