This achievement is pretty simple, yet very time consuming. I would say you shouldn't worry about it at least until you complete the 1000 Play With Others missions.
The easiest way to explore an area is hop onto your Hoverboard (holding
), and go around the edge of the map. This will get you at least 75% exploration, then you have to start going through the center of the map, along the way a message pops up that you explored the map.
Sometimes the message doesn't pop up, so if you're not sure whether you explored the map or not, press
, and go into Objectives -> Badges.
Side note 1: The achievement description is kinda misleading, because it says you have to FULLY explore the map, although you only have to earn the Exploration badge and successfully complete the current mission.
Side note 2: If you've already completed the 1,000 play with others, 10,000 survivors and the 500,000 building, and you really want to rush this achievement, go into a low level Destroy The Encampments mission and rush through it, this way you save a lot of time, instead of doing 20 minute missions.