[Automatic Cloud Backups and Sync is currently in development, and due to be released shortly]
At the moment, Export/Backup must be performed manually within the app.
From the main app menu, select "Log Book Backups".
This will present a new screen, where you can tap " Export". This will then save all the data within the app to a specific file (location will be shown on the screen).
You can then store this file for backups, or transfer to a new device.