The phrase originates from a real historical event. In 1715, the British government issued a proclamation known as the "Riot Act." This Act stated that if a group of 12 or more people were assembled unlawfully and refused to disperse when ordered by a magistrate, they could be deemed rioters and subject to arrest and punishment.
Over time, the phrase "read the Riot Act" came to symbolize the act of imposing order and enforcing discipline. Therefore, when someone reads another person the Riot Act, it suggests that they are being given a stern warning and told in no uncertain terms that their behavior is unacceptable.