Why was it such a revolutionary action for a black person to sit down at a "whites-only" luch counter in 1960 North Carolina?

Because of the policy of Segregation, it was unheard of for a black person to sit in a whites only area in 1960 North Carolina.

Explanation:

Although USA's constitution guaranteed equal rights for all citizen in principle, segregation had continued into the 1950s America as the Black population still struggled for complete civil rights.

Civil Rights Movement was built on the backdrop of many such instances of defiance of the Segregation.

Sitting in a white only lunch counter was the open defiance of a law that had been around for centuries and asserting a right of equality according to the constitution.