How can I record my iPhone's screen?
Since iOS 8 and OS X Yosemite, you can connect the iPhone to your Mac using the Lightning connector, and it will be available as a video camera. Then you can record video of what happens on the screen using QuickTime Player. 

(Source: https://developer.apple.com/app-store/app-previews/)

With previous versions of iOS and OS X, you can use AirPlay mirroring on your phone and  an AirPlay receiver/recorder like Reflector or X-Mirage.