After you have repaired the elevator generator using repair grenades, the next objective will be called 'Get to the Purification Plant'. Follow the arrows (press the 'Back' button for these to appear) and eventually you will see a cutscene.
Now you will have to take down the water tower, either the Rocket Launcher or Charge Launcher work best for this. Once this is done kill all of the enemies in the area. Remember to use Shockwave if you are overwhelmed. After this, pick up the salvage left behind, then hold X to return to Bastion. Once done the achievement will unlock.