Campfire mode triggers at the start of a player’s turn during the Onslaught Phase when there are no opposing faction units in play. This includes both opposing minions and heroes (but not does not include landmark units). Campfire mode is only intended to be triggered if one side has completely wiped out the forces of all of their opponents.
This is not the same as the trigger for the end of the wave, which happens when there are no faction minions left in play on either side. The wave can end without campfire mode ever being triggered.