This is a quest that forms part of the Minutemen story arc. Once you've met Preston Garvey in Concord at the start of the game, he will move to Sanctuary after you help him. When you meet him there, he will give you quests to help out other settlements. Once you have completed enough of these, he will ask for help taking back The Castle. This begins Taking Independence quest.
You start by meeting Preston and several other Minutemen at The Castle and after a brief discussion, You'll need to clear out some mirelurk eggs, after which a giant Mirelurk Queen will attack you and the other Minutemen. I've found the best way is with a Fatman and a couple of mini nukes. After the queen is dead, take out the leftover eggs.
Once the eggs and queen are dealt with, you'll need to set up a radio transmitter and one large or two medium generators to power it, which require that you have the Science Perk.
Once the radio tower is built and hooked up to the generators, just talk to Preston, the quest will complete and the achievement is yours.