Let's say you have 1,000 fans on your Facebook page. If you want to convert them into Messenger subscribers, all you have to do is give your fans a reason to message you (i.e. offer them something they can't already get by just being a fan), then let them know about your offer.
Right now you can expect 2% of your Facebook fan page audience to become Messenger subscribers, as long as you give them a good reason to message you.
That means if you have 1,000 fans, you could get 20 Messenger subscribers
With 20 Messenger subscribers, you could send messages that 16 of them read and that four of them click on.
Compare that to email, where if you sent the same email to the same 20 people, MAYBE 1 of them would read, and probably zero will click.
As soon as somebody messages your page, you can message them back, because they're your subscriber!
Hope that helps!