hi

Sorry the ONLY bluetooth device a 3rd generation Apple TV supports is a bluetooth keyboard. If you want to use your headphones you need to upgrade to an Apple TV HD or Apple TV 4K.

I have both Apple TV 3rd generations and a Apple TV 4K, the 4K can now support 2 pair of AirPods or AirPods Pro simultaneously.

What is your model Apple TV? Identify your Apple TV model - Apple Support

Bluetooth headphones are supported for the Apple TV (4th generation or later).

To pair bluetooth devices: Use Bluetooth accessories with your Apple TV - Apple Support

Verify that your headphones are in pairing mode (i.e. discoverable and ready for pairing). There is often a button for that.

I'd recommend the 4K model, if you have or are planning in the future to get a 4K TV then you will be all set. In addition the 4K Apple TV's are a HUGE upgrade over your current model.