Q: How to get the AirPlay instruction screen to go away from my screensaver?

Hello cogitisimo,

Thanks for using Apple Support Communities.

If you are seeing a display on your screensaver with instructions on how to connect to AirPlay, then you have the Conference Room Display option enabled in your Airplay settings on the Apple TV.  

Apple TV (2nd and 3rd generation): Understanding AirPlay settings - Apple Support

I'm not sure why you're trying to pair your Apple TV with your iPhone, as the only function for Bluetooth is to do the inital setup of your Apple TV.  Are you trying to get the Remote app configured?

Apple TV (3rd generation): Using an iOS device to set up your Apple TV - Apple Support

Set up Remote app for iPhone, iPad, and iPod touch - Apple Support

Take care,

Alex H.