ETJ stands for "Extra-Territorial Jurisdiction" and is an area outside the City's corporate limits in which the City can exercise land use and zoning regulations. Statewide authority for municipal extraterritorial land use regulation was granted by the State Legislature in 1959. The maximum size of a city’s ETJ area is determined by its population. Brevard's extraterritorial area may extend up to one mile from the city limits since its population is less than 10,000, however it does not have to extend the full mile.
For more information, you can read the legal summary "Extraterritorial Jurisdiction for Planning and Development Regulation" from the UNC School of Government.