What was the supreme court's reasoning in brown v. board of education? how important was the court's ruling in advancing civil rights?

In 1954, sixty years after Plessy v. Ferguson, the Supreme Court ruled unanimously in Brown v. Board of Education that “separate but equal" was unconstitutional under the Equal Protection Clause of the Fourteenth Amendment.
In a nutshell, the court decided the case, which is still legally controversial, at least among conservatives and libertarians, on the basis of its conclusion that "separate but equal" is inherently unequal. That seems to me to be a pretty unremarkable conclusion; I think the claim that institutionalized segregation doesn't disadvantage the minority is a very tough sell.
In terms of law and policy, I'd say Brown pretty much created the foundation for the civil rights movement, by making clear that in public life segregation was simply not an acceptable approach to racial and ethnic differences, and that nobody should be required to sit in the back of the bus.