The story never explains how they get the keys, which was the part I was the most confused about. But after looking into it more, what I think is happening is that they tell the person to choose any home on the Rently website. I have used Rently before, which has an awesome feature that lets you tour any home on your own schedule (without a real estate agent or homeowner). You basically just schedule an appointment through the app, then at your scheduled time you go to the home and you get a temporary code for a lockbox which has the key inside. You let yourself into the home and tour it, then put the key back when you are done.
I am assuming that after you "choose" the home on Rently, the people running this Instagram account put fake IDs and info into the Rently form, and then show up and get the keys from the lockbox. They probably cut the lockbox off after that (or maybe they don't care that much and just leave it on). Then they bring you the keys. Then you take the keys and squat in the house.
Which is funny, because there's no reason you need the middleman here. If you're going to be a criminal by squatting you might as well save the $1,400 and steal the keys yourself.