Increasing your following on Twitter and engaging that following are not exactly easy tasks. One quick tip to engage all new followers is to personally respond to their following you with a direct message. The direct message should thank them for following you but should also give them some kind of call to action.
The CTA could be a link to your website, latest blog post, free ebook, or some other giveaway. Whatever your CTA is, you want to engage with them outside of Twitter as well. The only trick is to respond back to the new follower ASAP, and give them a call to action they will actually want to click on.