To stream you need to send an RTMP stream to a Twitch Ingest server.
To get the users Stream Key you use the Get Stream Key endpoint.
To get the list of ingest servers you use the Get Ingest Server endpoint.
And as for getting the user to send the RTMP stream to that destination, that’s outside the scope of the Twitch API as it’s for you to develop and/or integrate with broadcast software.