One of the fastest ways to get a lot of followers is to join a reddit train, or another online group that has people follow one another. There may be active and inactive users who follow you, but just having large numbers helps a lot for the legitimate people whom you want to follow you to click the button to do that. When they see that you have a lot of followers, the first thing they think is “I want to follow this user because they have a lot of popularity, and must be sharing things that are important or well-liked, and that means I could benefit from that”. Just like selling itself, there is a psychological aspect to it.
Another fast (but temporary) way to get a lot of followers is to hold a raffle of some kind, to where only the people who follow you are eligible to win. By having them follow you, they will stick around to see if they’re a winner. Since entry is free, all they have to do is follow (and people love free stuff, so they will follow you). Alas, once the raffle is over and the winner is announced, there will be a large percentage of unfollowers. You’ll still have a lot of people who will stay, either because they like your tweets, or because they hadn’t seen that the raffle was over.
These are just a few methods, but they work.
(Note: I wanted to add this to my original response, but it seems that the ability to edit posts disappears after a certain amount of time??)