GrayHaze is a live streaming service built on top of ATProto. ATProto is an open source protocol for building decentralized social applications. It defines how to identify a user, where to put that users data, and what the user can do to move their data around. User data is stored in a PDS (Personal Data Store), which in most cases is owned by BlueSky PBLLC, however with a little bit of tech savvy you can transfer it elsewhere or even self host it. The dominant service that uses ATProto is BlueSky, a short form text based social media platform. GrayHaze takes the protocol and applies it to live streaming. While streaming, video segments are uploaded to your PDS and then served out to other users. Similarly, chat messages, and most other data that you produce while using the platform is stored on your PDS. The only exception to this is the authentication token used to log in to the site and perform necessary PDS operations like for sending chat messages.
The easiest method is to sign up for BlueSky. You will be given a handle that you can then plug in the GrayHaze log in input box. There are other more difficult methods, like hosting your own PDS and identity, or finding a PDS that's accepting open registrations, but realistically those are too much trouble for someone that's just trying to get in and yap at a streamer.