With the device powered on, press and hold down the Volume Down button and Power button together for one second. The screen will flash briefly and a smaller image of the screen will appear in the center of the screen to indicate a successful screenshot. If the volume is turned up, you'll hear a shutter sound just before the screen flashes.
I really wish Amazon would allow the "Alexa, take a screenshot" command (to which it responds) to actually do that. The last time I checked, it just opens the camera app. I sent feedback via the Alexa app suggesting the change, but it wouldn't hurt for more people to do the same.