One of the best ways to help grow your email list quickly is to have a giveaway and give people something for subscribing. Usually digital products will be best for this in most cases. You can offer and eBook, report or some other kind of digital items that people can receive for subscribing.
Once you have figured out what you will be offering then you will need to promote it. The social media websites work very well for this as a free option or you can use PPC advertising if you are looking to go the paid route, which can be quicker.
It all comes down to giving people something in return for subscribing and a good reason to subscribe. Everyone wants something from someone so give them something they want and they will hopefully give you what you want.