In order to take over territory, you need to complete side missions.
Side missions are the yellow icons on your map. Sometimes they are immediately present (typically Satellite Uplink missions, which will make more missions visible on your map
), sometimes you will see a mission randomly pop up on your mini-map, and sometimes you need to drop an enemy to make a mission appear (usually with a package interception mission).
Basically, just run around in the red areas of the map and, if no mission is currently available, it will become available fairly quickly. Areas you take over have a yellow tint, and after every mission you will get an update on what percentage of that territory you have claimed. Obviously, you need 100% in order for the trophy to pop in the respective district.
THE ONLY EXCEPTION TO THIS RULE is the last area in the Historic District - in order to obtain this trophy, you need to beat Kessler. After that, the final areas in red will be yours (it looks like 2 areas
). If you complete all the side missions in the Historic district AND see when you save that you have every territory claimed, do not freak out or think the trophy glitched - just beat the game. It will pop after the credits roll and you are back in the game.
Questions, corrections and neg vote justifications below.