Tap on the icon of the white box at the top left corner. See image above.
Choose Audiobooks or Podcasts to select the library you want to view.
Make sure and refresh the library first (Takes about a minute)
The Books tab at the bottom is the complete list of books, but you can choose the Authors or Collections tabs as well.
You can sort the books by Title, Author, Recently Added, or a bunch of other filters.
You can also just search at the top of the app on the Home tab.
If you find one just go into it and tap the Start Reading button. It will save your progress even if the app gets closed. It works with CarPlay also.
To avoid the $5.99 premium fee you will have to stream the audiobook or podcast from your device. The premium fee lets you download it to the device for offline playback.