There are two ways to access the library. The website or a mobile/desktop app. Both give you the same library and your progress should sync between them automatically but has been buggy in for me in the past. YMMV.
Open any browser and go to books.rosebromedia.org. No install required.
The apps should all support offline downloads and background playback. Check the Apps page for a few options.
Both website or all sign in with the same username and password. The app asks for a server address which is the just the website link.
https://books.rosebromedia.org
The ABS official app does not need to include https, however I cannot say for other apps.
assets/login-screenshot.png
The AudioBookShelf official app lets you download books directly to your device so you can listen offline. I'm sure most others do as well.