Make sure you have not only the Kindle app installed on your mobile device, but also the Audible app. It's the latter that will handle the heavy lifting, audio-wise. Update: According to Audible, the Kindle app has the necessary audio tech built in, and therefore the app itself is not required.
The best way to see this in action is with a book that supports Whispersync for Voice. You can use Amazon's Matchmaker service
to see which Kindle books you already own are able to be upgraded with audio, but you can also start with a freebie: Amazon is currently offering the classic "Black Beauty" for Kindle at no charge
, with accompanying professional narration also at no charge. (After you "buy" the book, you'll see a button allowing you to "buy" the narration as well.)
Load the Kindle app and open "Black Beauty." Tap anywhere on the screen to bring up the options menu, then look in the lower-left corner. You should see a headphone icon. Tap it, then tap the blue download arrow on the subsequent screen. This will start the download of the book's audio companion file.
When the download is done, you'll see a familiar-looking set of play controls. Tap the big blue triangle to play/pause, or either of the quick-skip buttons to go forward or back in 30-second increments.
To return to the text version, tap the little book icon in the lower-left corner of the screen. Because the book and audio editions stay in sync, you'll always be at the right location as you jump back and forth.