Full screen mode enables you to watch your stream with no further display content. In order to enable full screen on your iPhone, you need to rotate your device to landscape.
Additionally you can also zoom the picture by double tapping on the screen.
IMPORTANT - especially if fullscreen does not work:
Please check whether the portrait orientation lock is off. Otherwise full screen cannot be enabled. You can deactivate the portrait orientation lock in the control center of your device. From any display, swipe up to open the control center. If the portrait orientation lock button is activated, please deactivate it by tapping on the icon. Once you are done, try again. Zattoo should be streaming in full screen now when you rotate your device to landscape.