The Apple TV (1st generation), the 7.7 inches big silver-white slab from 2007 doesn’t support AirPlay.
If your Apple TV box is 3.9 inches and black, then it is at least 2nd generation.

Identify your Apple TV model - Apple Support
AirPlay is a function that has to be activated on the Apple TV box (in Settings﹥AirPlay﹥).
AirPlay may have restrictions as set by the content provider. They may require a newer model for it, depending from which app you wanted to stream.
Use AirPlay to stream video or mirror the screen of your iPhone, iPad, or iPod touch - Apple Support
If screen mirroring or streaming isn't working on your AirPlay-compatible device - Apple Support
The serial number and model number is also mentioned on-screen, in Settings﹥General﹥About﹥.
While there, also verify the Activation status. If that says Failed, then that has to be corrected before the unit can be used for AirPlay.