Sounds like you are GETing and showing the Twitch Login page instead of redirecting the user to Twitch?
The following:
GET https://id.twitch.tv/oauth2/authorize
?client_id=<your client ID>
&redirect_uri=<your registered redirect URI>
&response_type=<type>
&scope=<space-separated list of scopes>
Should read:
REDIRECT THE USER TO https://id.twitch.tv/oauth2/authorize
?client_id=<your client ID>
&redirect_uri=<your registered redirect URI>
&response_type=<type>
&scope=<space-separated list of scopes>