At launch, you can only play Google Stadia on TV by using the Chromecast Ultra included in the Founders and Premiere Edition bundles. But some users who already have Chromecast Ultras have been able to stream games through a firmware update. Google had plans to update older Chromecast Ultra models, but they haven't given a schedule for its release.
According to a Reddit thread, some users tried heading into the Settings and turning on the Preview Program and then rebooting the Chromecast to get the firmware update and play games. Google confirmed on Thursday that it will begin shipping early next week. The manual firmware update could be a workaround for those who got the access codes but don't have their bundles yet.
Source: Android Police