The payment feature, called Super Follows, will allow Twitter users to charge followers and give them access to extra content. That could be bonus tweets, access to a community group, subscription to a newsletter, or a badge indicating your support. In a mockup screenshot, Twitter showed an example where a user charges $4.99 per month to receive a series of perks. Twitter sees it as a way to let creators and publishers get paid directly by their fans.
This seems like it will be great for artists and such, but I cannot think of one person whose thoughts I care enough about to pay for. It will be interesting to see how the feature becomes utilized.