Do I need an Internet connection in order to listen to an audiobook?

Yes, but... The need for an Internet connection varies depending on use.

To read and listen online, a continuous Internet connection is required, both on a computer and in a mobile application. Streaming downloads and temporarily stores on your device a file containing a part of the book (e.g. a chapter); so offline playback is possible for a while, but will be interrupted at the end of the downloaded file if no connection is available. It is therefore possible to keep reading offline, but only for a short while.
To download an audiobook, a connection is required at the time of download only. Once downloaded, the audiobook remains accessible offline in its entirety.
To delete a downloaded audiobook from your device, a connection is required at the time of deletion, so that your playback position can be saved.