John Voorhees, on Airfoil Satellite TV:
The app, which can receive an audio stream from Airfoil for macOS, is available as a free download on the Apple TV App Store. When you open Airfoil Satellite TV on your Apple TV, a new audio destination appears in Airfoil on your Mac named ‘Airfoil Satellite on [Your Apple TV Name].’ Pick that destination and music starts streaming from your Mac to your Apple TV.
Airfoil for the Mac was able to send audio to the Apple TV until last month when tvOS 10.2 broke compatibility. With version 10.2, tvOS requires any device attempting to send media over AirPlay to authenticate itself with an automatically generated on-screen passcode or a user-created password. This functionality is already built into Apple’s software, but third-party apps like Airfoil are being forced to find workarounds.
The good news is that Rogue Amoeba, the developers of Airfoil, are hard at work on an update that will allow you to stream audio to your Apple TV without the use of Airfoil Satellite TV. They haven’t revealed how this will work in detail, but my guess is that they’ve found a way to authenticate Airfoil with the Apple TV using a user-created password in the AirPlay settings.