The Brief: DansGame is a Twitch emote used to represent annoyance or disgust.
DansGame is a Twitch emote. It represents anger, disgust, or annoyance.
— trav (@trav1099) February 9, 2016
The emote is of Twitch streamer DansGaming making a face of pure disdain. According to Dictionary.com it dates back to 2010. If streamers are unable to use the DansGame image, they may write out “DansGame” instead.