The Connect to Twitch button goes to Twitch:
https://api.twitch.tv/kraken/oauth2/authorize?response_type=token&client_id=MY_CLIENT_IDf&redirect_uri=REDIRECT_URI
Then, Twitch, after they authenticate the user for you, automagically push the user to REDIRECT_URI. Whatever that may be (http://nowhere.com/twitch_landing.html – whatever).
Cheers