Hi,
Your stream is twitch.tv/thatavsgamer? I couldn’t see the twitter card meta tags on that page.
It should just be a case that once the meta tags are in place on your page you get it approved via the twitter cards validator and when you are white-listed you should be good to go, paste that twitch.tv/thatavsgamer into a tweet and it should be picket up.
Note that it appears you to have to use the “Show Preview” link on a tweet to actually get it expanded to show the contents (e.g. your image), not sure if this is true for all cards.