Ecamm Live will automatically show a connected iPhone or iPad's screen as a camera source when the device is connected via USB.
If this isn't happening, the device may need to be paired with the Mac.
Pairing an iOS Device in macOS 10.15 Catalina
In macOS 10.15 Catalina, iOS devices are managed in the Finder. Look for the device in the left-hand sidebar of any Finder window. Click it and follow the steps to Trust the device. (If you see device settings instead, it's already trusted.)
Meanwhile, on the iOS device's screen, you' may see a message as well.
Pairing an iOS Device in macOS 10.12, 10.13 and 10.14
In macOS 10.12 (Sierra), 10.13 (High Sierra) and 10.14 (Mojave), devices are managed in iTunes. Open iTunes, and click the tiny phone icon above the sidebar. Look for this message.