To save memory on your device, all your purchased audiobooks are stored in the cloud by default. To download an audiobook to your device, simply tap on the cover and select “Download Audiobook” You’ll then get a progress bar showing the status of the download. You can tell which audiobooks are stored in the cloud by the cloud icon on the cover. You can also use the filter bar at the top of your library to show all downloaded audiobooks.
Once downloaded, tap on the cover to open the play screen and then the play button to begin listening to your audiobook.