This achievement is unlocked by neutralizing the seven amplifiers in the city of Insomnia. Once the seventh amplifier is neutralized the achievement will pop.
The mission is clearly explained in the story including how to initiate the fight at each amplifier. You must neutralize three amplifiers for the story to progress thus taking over three sectors in Insomnia. The story can progress after that so the achievement is missable if you do not stay in Insomnia to take over the four remaining sectors.
All amplifiers are clearly indicated on the map. In addition, a purple beam shoots in the sky from each amplifier. Moreover, an indicator is provided showing the distance to the amplifier Ardyn is targeting. There are forces on the ground but they have no bearing on taking over sectors. The only condition to take over the sector is to neutralize the clearly marked amplifier in that sector.
The fights at each amplifier become more challenging the more sectors are taken over. You can buy curatives and items that buff power, defense, magic, and sprit to assist in the fights. Moreover, summoning Ifrit especially with Meteor Strike can greatly simplify the challenge. After the seventh amplifier is neutralized taking over the seventh and final sector, the achievement will unlock.