The Old Way: You send people a friend request on Facebook. If they approve it, you get to see their stream of data.
The New Way: You can subscribe to someone - if they've opted in to subscribe. Whether or not they approve you, you'll get to see the info they've made public.
Facebook is emphasizing the fact that you have to opt in to get the button on your profile.
The “Subscriptions” feature...addresses one of the biggest pain points for Facebook users who have a lot of friends... It means that power users who want to follow the updates of public figures now have a way to do so, without having to submit a friend request to the person in question. It also provides a way for users to breach the previously constricting 5,000 friend limit on Facebook.
There are a lot of details about how it works here.