Find the anime on the phone, press Watch now, pick the service, and use the cast icon in that service's player: Chromecast from Android, AirPlay from an iPhone, the built-in cast on Samsung and LG. The stream goes to the television in HD with the same audio menu.
Television browsers
Android TV, Google TV and Fire TV's browser open the site directly and the tabs work with the remote; Watch now opens the anime in the service's TV app. On a PC, the app installs from the address bar.