I've used "Easy Outs" my entire life to get out broken studs and snapped
off bolts. I guess I have learned something new, but don't really
understand the "why."

I've never had much luck with "Easy Outs". Mostly it was just a case of the
bolt being too stubborn to move though.

When I've used the Easy Out (extractors), they were always tapered and
had left hand threads. I just drilled an appropriately sized hole in the
bolt or stud, inserted the Easy Out into the hole and began "tightening"
it until the bolt came out. What would be the reason or advantage of
drilling the hole with a left hand bit?

The big advantage is that the drilling action may loosen up the bolt and
make it easier to remove.

Could the reason be that using a right handed bit on a
LOOSE-but-broken-off stud or bolt would move it further into the hole as
you drilled? I've never had one that was loose, so I never worried about
it. In all my cases, the blasted things were frozen solid in the hole. Is
that the reason for the LH drill bits?