Under "My library" in the app menu, you will find "Want to read". Here you will find all the books you save for later. You save books by clicking on the heart icon next to the books you find interesting in the app.
A book that you start reading or listening to is saved in the "Reading now" tab. The book is also downloaded to your device so you can read or listen offline. You can change the download settings in your profile at any time. You can find the settings under the profile page under the heading "Download".