The Twitch website does the following:
POST https://api.twitch.tv/api/users/:name/follows/games/follow
POST https://api.twitch.tv/api/users/:name/follows/games/unfollow
with name=Game Name Here as the body of POST requests.
From what I can see, these require you to have valid session and API token. I’m not sure how the mobile clients are able do it using just the Twitch API, as I cannot seem to get the right scope to be able to (you would assume user_follows_edit would work, but it doesn’t appear to).