Why were ordinary people willing to take part in the civil rights movement?

Depends on what you mean by ordinary - some ordinary folks were directly affected by discriminatory state laws and practices (some would say they still are), some ordinary folks felt as though such laws and practices were simply un-American (some would say they still do). There are some folks who veered towards violence which most people probably agree is not ordinary. My reading of history shows there was much, much more violence directed towards the civil rights activists in the 1960's than the other way around, but there are certainly examples of violence and criminal activity attached to the Nation of Islam and the Black Panther movement.