When your Apple Watch is paired with your iPhone and the initial setup has been completed correctly, Pillow will automatically track your sleep without requiring any further action on your part.
Simply wear your Apple Watch to bed for the duration of your sleep session and Pillow will take care of the rest. In most cases, no special calibration is needed, but if you notice any inaccuracies, a quick calibration may be required to improve results.
To learn how to calibrate your Apple Watch, you can follow Apple's official calibration guide