You could potentially wait for the Game Engine Plugins to be made available in an Open Beta (or release) State which could cut down on your need to implement many things if you are using Unity or UE.
Otherwise, what Barry explained remains true and you’ll want to open a browser → user logs in → token gets relayed to game.
As an Example, I can provide my (no longer functional with the current game version) entry for the Channel Points Hackathon, which is a Mod for a Unity Game which follows the exact steps mentioned above. GitHub - Marenthyu/SCPI: BTD6 Streaming Channel Points Integration
More resources can also be found here: Authenticating with the Twitch API or on the Discord