The process is explained in the announcement post: https://discuss.dev.twitch.com/t/introducing-the-game-developer-global-chat-badge/40033
Essentially, to be eligible for the Game Developer badge you need to be part of an Organisation on the Twitch Dev console, and that Organisation must have claimed ownership of a game category that is on Twitch.
It is not possible to apply for the Game Developer badge directly, instead if you are currently working for a Games Studio you would need to reach out to them and have them add you to their organisation on Twitch. It’s up to those organisations to determine if they wish to enable the Game Developer badge and which staff they wish to have as part of the organisation.