An audio book club can be a group of people who only have time, or just prefer, to listen to books instead of reading them. It could also be a subscription to an audio book service, such as Audible, AudioBookWorm, Simply Audiobooks, or etcetera. Which service to choose can be a difficult decision, but with the low price of online distribution and frequent trial offers, it's not unreasonable to simply try different audiobook clubs until you find the one for you.
There are multiple ways to find an audio book club in your area. Try calling or visiting your local library to see if they are familiar with an already existing audiobook club. They may even host one themselves.
Check online as well. It may require a little searching, but in larger cities, local book clubs may have websites with contact information and how to sign up. Finally, there are forums online where people all over the world discuss ebooks and audio books, from dedicated forum sections to single threads on related writing sites such as NaNoWriMo.