What is the fundamental idea behind the fourteenth amendment to the constitution, which is mentioned in Brown v. Board of Education?

The Fourteenth Amendment secures the rights of citizenship to all Americans.

Explanation:

The fundamental idea of the Fourteenth Amendment is that it secures the rights of citizenship to all Americans, including African Americans: It extends citizenship to all "persons born or naturalized in the United States, and subject to the jurisdiction thereof", protect the privileges and immunities of U. S. citizens, and guarantees due process of laws and the equal protection of the laws to all citizens.

This idea was also mentioned in the landmark civil rights case "Brown v. Board of Education" which outlawed segregation in American, alleging that segregated facilities were not equal and instead, they were a form of discrimination that instilled a sense of inferiority to African Americans children; and thus it violated the right of equal protection of the laws to all citizens, protected in the 14th Amendment.

The Fourteenth Amendment secures the rights of citizenship to all Americans.