Is there a particular reason for the way that החצר seems to be pronounced here hechatser (rather than hachatser). An interaction between two successive 'a' vowels, or something more particular than that ?
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 agree with radagastthebrown. Learn these rules only if you have an academic interest. Nobody speaks like that unless they are reciting poetry or narrating a state ceremony, native speakers learn these rules at school and many of us forget them later.