I think the Wikipedia page summarizes it quite well; the intent was to oppose racial integration of public spaces.
Fundamentally it was a disagreement about segregation and integration. The authors of the Manifesto wanted to preserve a specific culture of segregation and privilege. The Supreme court required integration, which was a threat to that culture. From before the founding of the country, we have been aware of the "peculiar problem" of the South, and aware that it was an existential problem. The Constitutional Convention agreed to not discuss it, and agreed to the 3/5 compromise as the price of nationhood, but it just deferred the problem. The Missouri compromise deferred the problem. The Civil War decided the problem, and we're still carrying out the implementation of that solution.
Segregation, diversity and Supreme Court rulings touch on Federalism vs sovereignty, autonomy vs central power (in this context, I wouldn't call it "Federal" power) and a host of other things. But at the core the Southern Manifesto was an attempt to preserve a system that we as a country have determined to be racist and unacceptable.