I have no idea what mantra he uses. The recording seems to indicate a guided meditation with a mantra. It is hard to imagine how this works - there are so many things going on, that the body and the mind will not quiet down for most of the people.
It is easier to pick up one mantra (like I AM), and use it to go in deep meditation. If you read lesson 13 (http://www.aypsite.org/13.html), you get all the instructions you need for a meditation with the mantra. The results depend on the consistency of your practice - so meditate twice a day, for 20 minutes. Best wishes!
My first impulse when I listened to the recording was to say that the mantra in the background was OM... It might be - it might not be... I thought I should mention it to you.