Large Scale lightning Strikes from Cloudless or Near Coudless Skies. How Does it Work?
Revised answer

TL;DR After having just heard from two experts who actually work in this field(firestorms and dry lightning) in Australia I believe the lightning seen is what is known as "dry lightning" comes from isolated thunderstorms and sometimes storms associated with minor troughs and fronts that come through Australia and also originating from a pyrocumulonimbus cloud. So @David Hammen and @jamesql are correct in their observations that the large scale dry lightning is indeed originating from isolated thunderstorms as well as storms associated with minor troughs and fronts. This dry lightning the experts confirm  is associated with little rain. 

In addition there have also been several  dry lightning strikes from pyrocumulonimbus clouds. As an example for today over 20 dry lightning strikes have been recorded originating from pyrocumulonimbus clouds. 

The major driver for the current drought like conditions over Australia is the large anomalous positive Indian Ocean Dipole (positive IOD) that has persisted well into boreal autumn/astral spring. 

What the experts mention is that while the dry hot conditions are sporadic (2-4 days) they are persisting over over many weeks.

The following articles written by two researchers  Firestorms and flaming tornadoes: how bushfires create their own ferocious weather systems
 and Firestorms: the bushfire/thunderstorm hybrids we urgently need to understand provide information on the dynamics behind firestorms and the formation of pyrocumulus clouds and the associated dry lightning. 

Following is a  summary of those articles and the conditions that lead to dry lightning from pyrocumulonimbus clouds.

Not all bushfires lead to firestorms. If a bushfire has sufficient area the upward movement of air causes the fire to interact with the atmosphere above it and form what is known as a pyrocloud. If in addition  there is an atmospheric instability then this process leads to the formation of a pyrocumulonimbus cloud . Normally in meteorology we are taught of heat radiating from the ground but in this case the upward movement of air is due to the heat emanating from the fire. 

Again from mesoscale meteorology one hears the word "downdrafts". Similarly in this context the appropriate word is "downbursts" and these are vertical drafts of air that hit the ground and move about in all directions. The unsettled conditions cause embers to carry over large distances. 

These firestorms produce dry lightning that potentially can spark new fires that may end up creating a larger flaming zone.