Update: According to Audible, the Kindle app has the necessary audio tech built in, and therefore the app itself is not required. It's the latter that will handle the heavy lifting, audio-wise.
Step 1: Make sure you have not only the Kindle app installed on your mobile device, but also the Audible app. Note that my testing was performed on an iPhone, but the process is virtually identical in Android. However, the process of doing so isn't immediately clear, so let's take a look at how Whispersync for Voice works in Kindle apps.
You can be listening to a book in the car, then switch to the text when you get home and want to keep reading.
Some of the books you already own can be upgraded with narration.