What does the iCloud backup save, and how do I restore it?

It saves what pon has learned on this device (order, locations, pantry, history) — in your own iCloud, which only you can access; we can't reach it. In the app settings (profile icon at the top right of the overview) → „Backup“ you can back up, and via „Restore from iCloud“ you can restore: „Merge“ (recommended when there's already data on the device) or „Replace“ (a fresh device switch).

Give it a try: that way, when you switch devices, nothing pon has learned about your shopping is lost. Your lists themselves come back through normal account sync anyway, not through the backup.