Having written on this before, I'll repost that here.

Most basically, it wasn't actually a war. It just makes for an amusing description of what was a pest eradication campaign as part of an inter-war period of settlement in Western Australia. Several thousand ex-soldiers had settled there post-WWI in hopes of farming, but fluctuating wheat prices led to marked instability. Several years of political wrangling, bad markets, and agitating settlers were bad enough when, in 1932 an emu migration served to throw another wrench into the matter, the path that these thousands of critters were taking being through some of the wheat growing lands which were already struggling enough as it.

The farmers, unable to handle the matter on themselves due to the volume requested the calling up of local defense forces, who would be able to deploy machine guns, presumed to be much more effective than the assortment of rifles available to the farmers. A small detachment was sent off with two Lewis guns and 10,000 rounds of ammunition (which was hardly sufficient given reports of over 20,000 emus), and also accompanied by a move camera (this seems to be some footage from it).

As it turned out, they were not well equipped for dealing with emus, as they mostly ranged about in small groups rather than one clustered herds which Major G.P.W. Meredith had been planning for. A few dozen were killed in the first day after mounting several small ambushes at watering holes, but it was piecemeal and ineffective. The next day, when they finally found a much larger group, the machine-gun fulfilled expectations... until it jammed, only to be cleared after the survivors had fled. Attempts to pursue emus and fire from trucks proved futile as the bouncing terrain made it impossible to aim. One attempt to run over an emu was successful, but also took out the vehicle which lost control and crashed into a fence!

The poor showing by that point which had seen the expenditure of a good portion of their already insufficient ammunition was obviously quite embarrassing, and although Meredith claimed they had taken out 300 of the birds, it was likely a high estimate, and it didn't really matter since the press caught wind of it and was having a laugh riot. The Prime Minister, who knew nothing of the little endeavor, found himself being asked about the waste of funds in an apparently futile goose emu chase by Parliament, and the opposition started joking about how the emus were winning, which comes to the heart of your question.

In sum, it wasn't a war. It was no different, really, than a situation where the National Guard gets called up to assist with hurricane relief. The emus were considered a pest that needed to be dealt with, and a plan was formulated to do so. That plan however was bungled, and the description of it as "a war" came from people who were laughing at how ludicrous the entire endeavor was. One Parliamentarian, for example, asked the PM if "'a medal was to be struck for those taking part in this war", but he wasn't being serious, he was having a laugh at him! Others were more blunt and didn't bother with the jokes, such as the writer Alan Marshall who noted "[the emus] 'had made the defence authorities look like idiots".

Of course, after having ordered the machine-gunners withdrawn following the browbeating, the PM suddenly was assaulted by people asking for them to return! Sir James Mitchell, Premier of Western Australia, was insistent that the campaign continue. Major Meredith, his pride at stake, wrote a very puffed up report that only added to the hilarity of the matter, writing of the Emus as some noble opponent to explain the fairly low ratio of kills to ammunition shot:

If we had a military division with the bullet-carrying capacity of these birds, it would face any army in the world. They could face machine guns with the invulnerability of tanks. They are like Zulus, whom even dum dum bullets would not stop.

The Defence Secretary attempted to pawn off the matter to the Western Australian government, offering to loan them machine guns to be used by some qualified veteran located there, but none could be found, which meant "'Major Meredith and his merry men" were again sent out. Although perhaps a little more forewarned, nothing really changed. They continued to manage to take on small groups here and there, but with 'success' only meaning a few dozen birds a day, it was obviously a hopeless endeavor, and the emus, now catching on, became better at avoiding the group anyways, which highlights another important factor here, namely that this was a single group of men. Major Meredith, his two machine-gunners McMurray and O'Halloran, and a few additional support for the gun and driving. At best they were able to split into two groups, which they attempted at points, but that doesn't change their numbers.

This wasn't some war of maneuver, with hundreds of men in the field fighting valiantly against the dreaded emu foes. It was just a few guys in a pair of trucks going heater hunting, basically. They did get some help at points from locals, such as when the Merredin Rifle Club came to assist after a group of emus managed to penetrate through a gate that had been left unlocked on the rabbit-proof fence, but it didn't help all that much, harking back at best to the futile efforts of the farmers originally.

The final report tried to salvage some pride, nothing that although only "986" emus had been killed, roughly 5 percent of the reported number, Meredith claimed 2,500 had died later of their wounds, with no substantiation for this claim. Some 10,000 rounds of ammunition had been fired, and the cost was covered by the wheat-growers and other settlers in the region (or rather, the Agricultural Bank paid it and put liens on their farms). The eradication of course had not been all that successful, and the emus eventually left on their own accord to continue their migration. Similar migrations happened a few times over the next decade, but the farmers were basically left to deal with it themselves, although in 1943 they were given free ammunition from stockpiles deemed no longer likely to be used against a Japanese invasion.

As for the idea of an "Emu War", again, the entire farce struck a chord with the Australian public for its failure in execution, and became a symbol of government incompetence. They had hoped to show that they were doing something to help the poor, stricken wheat-farmers in Western Australia (who did genuinely appreciate the attempt for the most part), and instead failed utterly. Much of the settlement in the area in fact would be soon abandoned, although the plague of emus were only a part of what led to abandonment of farms in the region.

Sources

I was quite amazed that there was actually a decent paper on this, which does a nice job both covering the gist of the campaign, as well as situating it into the larger social situation of Western Australia at the time, pulling this out of meme territory and into a real little commentary. As such, much thanks to Murray Johnson on whom I mostly am drawing from here:

Murray Johnson (2006) ‘Feathered foes’: Soldier settlers and Western Australia's ‘Emu War’ of 1932, Journal of Australian Studies, 30:88, 147-157,