How to continue audio playback across Keynote slides

Ahh, the part that most people miss...

You need to add the specific audio track to each slide.

On each one, make sure the 'Play audio across slides' is enabled (it should be the default):

Now, as long as each subsequent slide has the same audio track on it as the previous slide, and the 'Play audio across slides' option is enabled, it will play continuously throughout your presentation.

If any slide does not have the audio track on it, the audio will stop at that slide, and restart from the beginning on the next slide that contains the audio track.