To use Cardboard Camera, you hold the camera as you turn around, similar to taking a panoramic photo with your phone. Google's processing converts the panorama to a 3D 360 image (albeit with limited vertical field of view). It's not a substitute for a real 360 camera, but it gives an idea of the benefits of 360 photos and 360 cameras.
If you like the results you're getting from Cardboard Camera, you might want to consider the Vuze (a 3D 360 camera for consumers), Insta360 Nano (a 360 camera for the iPhone), Ricoh Theta S, or the LG 360 Cam (review coming soon). You could also use the Samsung Gear 360, but it is not compatible with the iPhone (there are some workarounds).