Apple's official answer on this is a bit contradictory: their promotional page lists Apple Watch as a device you can use with them while their main support page only lists Mac and iPhone as compatible devices.
However, you can definitely use the Airpods with any Apple Watch running watchOS 3 or later. The Airpods should automagically connect to the Watch once you set them up on your phone using iCloud, but if you need to pair them to the watch manually, you can do it like you would with any bluetooth headset. Here's how:
Put your AirPods in the charging case.
Open the lid, but don’t remove AirPods yet.
Press and hold the circular setup button on the back near the bottom of the charging case until the status light between your AirPods at the top of the case starts rhythmically blinking white.
Launch Settings on Apple Watch, tap Bluetooth and select your AirPods when they appear on the list of nearby Bluetooth devices.
After device status changes from “Pairing” to “Paired,” AirPods are ready to use.
Start playing your music.
Those steps are from iDownloadBlog.
If it's still not working make sure:
- The Apple Watch is running watchOS 3 and is not in Airplane Mode
- The AirPods are within a few feet of the watch
- The Apple Watch isn't paired to any other Bluetooth device (other than your phone, of course)
- The AirPods are in pairing mode (press and hold the button on the AirPods' case)