The rule is a bit complicated, and isn't worth learning because no one speaks like that. Anyway, for it to be pronounced 'he-' instead of 'ha-', these conditions have to be met -
1) It has to come before either ע ,ה or ח
2) It has to have kamatz gadol (nikkud) ('ah' sound, but not every 'ah')
3) If it's ה or ע, the first syllable must not be stressed
But it's really not important in my opinion. I'm surprised they even use it in the audio.
I have nothing of substance to add to radagast & almogl. I'll just say that I was not surprised by the pronunciation because I started Hebrew with biblical Hebrew rules in which before an unaccented הָ and עָ and always before חָ nikkud for the definite article is הֶ It was interesting to learn from people who know Israeli Hebrew that that ancient rule (which goes back at least to the tenth century masoretes) is now relegated to poetry and such.