I was on the ANZ flight from Heathrow to Auckland and back
last year (flights NZ1 and NZ2)
As Keg has said, at that time, you did indeed get off the aircraft
at LAX, however you did not have to bother with collecting your
luggage.
After getting off the aircraft, you are directed to a transit
area (separate from the rest of the airport, you can't go
duty-free shopping or similar). You first go through the usual
immigration stuff (passport, form, fingerprints, etc), and then
everyone ends up in a transit lounge. Once the aircraft is
ready to go, you get back on the aircraft and continue to
Auckland.