Apple has added iOS screen recording to Yosemite and iOS 8. It is aimed at developers who want to make video demos of their apps, though it has some utility for other people as well.
- Plug your iOS 8 (or higher) iPhone/iPad/iPod Touch into your OS X Yosemite Mac by USB Lightning cable and unlock your device (sorry, this tip doesn’t work with older 30-pin devices).
- On your Mac load QuickTime Player.
- From QuickTime Player’s File menu, choose New Movie Recording (not New Screen Recording, as that will record your Mac’s screen, not your mobile device’s screen).
- On the recording button bar, to the right of the red circle, click the little down arrow.
- Switch the camera selection to your iPhone.
- Click the red circle record button when you are ready to start.
- When you are finished recording click the same button, which has now changed to a square inside a circle.
- Trim, export, import into a video editor, or upload as needed.
Unfortunately, it seems that QuickTime Player is the only program that can capture your device’s screen; other apps such as FaceTime can’t connect to it as a general purpose video device. Another shortcoming is that there is no visual indication of when or where you touch the screen, which would have been very handy for instructional video.