You sign up for an account with Audible. (Although if you've already got an Amazon account, that login will work on Audible too.)
You have to sign up to one of their subscriptions. Personally, I listen to loads of audiobooks so I pay about £110 annually which gets me 24 credits. In Audible World, 1 book = 1 credit.
Alternatively, the base subscription is £7.99 a month, and you get one credit per month.
Then if you want more books you can buy them. Most books are £7.99 each if you don't use a credit. Sometimes they have sales - usually either 2 for 1, or £3 sale.
You listen via the Audible app, on any device that you download that app to (and sign in). I pair my phone (I'm on Apple but it's similar on Android) with my bluetooth headphones, or the car's entertainment system, and listen through that - just like I would with Spotify or Apple Music.
As I understand it, you can't 'move' books from one device to another. Whilst I believe you can switch between devices, I don't think Android syncs very well; so if you get up to chapter 8 on your phone, but then go onto a tablet, it won't automatically find chapter 8 on the different device.
The Android version of the app is a bit clunky, I'll warn you now.
You can use credits to purchase books on the apps, but because of some convoluted argument between Amazon, Google and Apple you can't purchase via the apps at the moment (I don't think, anyway). I tend to buy on the desktop site anyway, because I tend to browse titles when I'm supposed to be working.