Some Twitter users are not seeing this new notification when they log into the app, alerting them to the new Pin Reply option.
As it sounds, tweet reply pinning will enable users to highlight a specific reply to their tweet via the three dots menu on a response. You'll then be able to keep that reply at the top of the thread, maximizing exposure for that specific interaction.
Which could be good for highlighting, say, celebrity engagement, or to help showcase especially helpful/engaging/funny responses to your tweets, which could also guide the relating conversation. There are also a range of potential brand uses, like inviting the most creative replies to your tweets, then pinning the best response.
Twitter's added a range of tools on this front over the past year, from reply controls to Communities to removing yourself from a tweet conversation. Twitter's working to give users even more capacity to manage exactly how they engage and interact within the app, and pinned replies would provide another small enhancement in this context.