To unlock this achievement you need to keep all six citizens alive in the optional quest "Save at least 3 citizens 6/3", which is under the War of the Worlds quest line.
Once the achievement pops for completing Act 1, you will come out from the underground via a hatch, and then the War of the Worlds quest line will begin, starting with "Defeat the Robots 0/3". Once you defeat those robots, you will head down to the village square for the "What's Happening in the Village?" quest. After a short cutscene, two more quests will pop up: the aforementioned "(Optional) Save at least 3 Citizens 6/3" quest, and the "Kill the Robots 0/6" quest. To unlock the achievement, the counter for the former must still be at 6/3 when the counter for the latter reaches 6/6.
Unlike the main game, where you can open the menu and manually save the game, the DLC only has autosaves. Luckily, the game autosaves right before this cutscene. So, if you fail to keep all 6 citizens alive, your best strategy is to let the robots kill you so that you can respawn and try again. If you accidentally defeat the robots, the game will auto-save and you'll need to start a new save from scratch and complete Act 1 again to get back to this point again.
Despite playing on Easy, I struggled with this for awhile because there were two to three times as many robots in the square compared to most video clips I watched of the achievement, such as this one from The Gamer Society (no affiliation to me)
It may have been a bug, but if it happens to you too, the extra robots were blue and could not be killed. You only need to kill the red ones (look for a red eye and a red waist). Move quickly from each cluster of robots around each citizen, killing just the red ones and keeping an eye on the 6/3 citizen and 0/6 robot counters to ensure all the citizens are still alive before you kill the final robot. Once you do, the achievement should pop.