The repair service (which currently only appears to be available in the US) requires you to enter the IMEI number of your device, at which point you can choose from various options to describe the problem that you’re experiencing with your phone. If your phone is still under warranty, then the IMEI number should be enough to let Google’s know to not charge you for the repair work. Any repair work should be completed within seven to ten working days.
The imminent launch of the Pixel 3 is likely to result in an influx of Pixel devices onto the second hand market, and they won’t all be in perfect condition. Dealing with third-party repair services can be a hit-and-miss experience, so it’s reassuring that Google is offering a repair service of its own.