Apple has made a bit of a change to the way audiobook purchases work through the iTunes Store, now allowing customers to re-download purchased audiobooks. The hasn't previously been possible, and it looks as though the change was made recently. Customers can download their purchased audiobooks both through the iBooks app on iPhone or iPad, and through iTunes on a Mac or PC.
According to an Apple support document, you can use the search feature in the iBooks app on iPhone and iPad to find audiobooks you have previously purchased. When your book appears in the results, tap the cloud icon to download it like you would for a song or video. In iTunes, you need only to search the store for the book, and tap the Download button.
It should be noted that this feature is not without its issues. For instance, if you purchased an audiobook which was later pulled from the store and then returned, you may not be able to download it. Such is the case with the Harry Potter series of audiobooks, which were available on iTunes before being pulled a couple of years ago. Though the books later returned, they are not downloadable for those who purchased them before their removal.
This is functionality that rival Amazon has offered for awhile through its Audible audiobook service, though it's nice to see Apple add this capability for audiobook lovers.