It's called a "setup" button, but it's really more like a "reset" button, and is used when you cannot successfully set up your AirPods via your iOS device:
Still can’t connect? Follow these steps on your iOS device.
- Go to Settings > Bluetooth and tap the icon next to your AirPods.1 Then tap Forget This Device.
- Put your AirPods in the case and close the lid. Wait 15 seconds, then open the lid.
- Press and hold the setup button on the back of the case until you see the status light flash amber a few times, then flash white.
- Place your AirPods close to your device. Follow the steps on your iOS device's screen.
From this support article: Get help with your AirPods - Apple Support
Cheers,
GB