Technicall, you take a screenshot on your Apple TV by pressing Volume Down + Play buttons simultaneously on the Siri Remote. But these screenshots are only used by Apple diagnostics and are not available to you.
To take a screenshot on the Apple TV that you can actually use, you need your Mac / Mac book on hand.
Assuming your Mac has at least MacOS High Sierra or later versions and at least a 4th gen Apple TV with tvOS 11 or later, the process is very simple.
Make sure the Mac and Apple TV are powered on and connected to the same Wireless network.
Then launch the Quick time player app on your Mac. On the Quicktime player app, click on File then select New Movie Recording. This will open the Mac’s webcam.
Click the small arrow to the right of the red button, then select Apple TV in the Camera section of the pop-up menu. Go through the pairing process and you’ll now see your Apple Tv screen on your Mac.
You can now take a screenshot on Mac like you would normally; this time though, you’ll capture the contents of your Apple TV screen.
For Older Apple TVs and Older Operating Systems
You can connect the Apple TV to your Mac using a USB-C cable. Then download Xcode.
Launch Xcode with the Apple TV turned on and connected to the Mac. Click Window > Devices in the menu bar in Xcode. Select Apple TV and click Take Screenshot.
The screenshot will be saved in your Macs default screenshot folder.