1. Relating to the part of a town or city where brothels and other commercial sex-based activities are concentrated.

From the red lights traditionally displayed in the doors and windows of brothels. Note: there is no explanation in the dictionary as to why they "displayed" the "red lights".

2. An area or district in a city in which many houses of prostitution are located [1890-95; allegedly so called because brothels displayed red lights].

At least in the U.S., some say the origin of the red light comes from the red lanterns carried by railway workers, which were left outside brothels when the workers entered, so that they could be quickly located when the trains were ready to leave.