Following a Buzz means you have opted in to receive notifications/alerts about that Buzz, for instance when someone posts a comment, or the Buzz is edited.
When you create a new Buzz yourself, you as the owner automatically follow it. When you create a Buzz using the New Conversation template, all participants are automatically set to Follow the Buzz so they will receive your messages - they can still manually opt out on the Buzz itself, or opt out directly from the notification emails.
In the app
Click the Buzz options button (the 3 dots at top right), and check or uncheck the 'Following' toggle on the Options screen.
In a browser
Simply click toggle the ‘Follow?’ button in the Buzz header. If you are following a Buzz, the same button switches to ‘Following’ - click it again to unfollow stop receiving notifications for that Buzz.