Brown was an extremely important case because it brought to light what was really happening in schools, and made people face the ugly reality about segregation: separate is not equal! This is an important realization to lead to the Civil Rights Act.
The Civil Rights Act of 1964 contained provisions that was intended to enforce the Court's decision in Brown. While the Supreme Court had ruled (in Brown II) that school districts should push to integrate "with all deliberate speed," the reality was that the vast majority (over 90 percent in some Southern states) of schools were still segregated. Title VI of the Civil Rights Act specifically created a uniform policy stating that schools must be non-discriminatory, and that the federal government should deny funding to institutions that discriminated.Title VII authorized the Attorney General to conduct lawsuits to bring about equality in education as well as in public accomodations. So while the first response is absolutely correct, it was connected in tangible ways, as well.
Brown goes with the Civil Rights Act because it sort of led to that act. It led to it in two sorts of ways. First, Brown led to it in policy terms. Brown was a case in which the Supreme Court ruled that segregation in schools was illegal. The Civil Rights Act of 1964 made segregation in many other areas (like restaurants and other public accomodations) illegal. Second, Brown led to it by helping to start the Civil Rights Movement. Brown helped to convince African Americans that they could actually gain rights. This helped cause the protests and such that led to the Civil Rights Act.