I don't understand how a Commonwealth country so removed from Europe paid such a high price.
They didnt.
The AWM gives the figures as circa 420,000 enlisted and 60,000 fatalies which is ~ 1.2% of the population. A report by the War Office called Statistics of the Military Effort of the British Empire During the Great War gives total Uk fatalities as about 700,000 which is ~ 1.7% of the population.
The official German history gives about 2,000,000 fatalities which is ~ 3.1% of the population. More than twice the Australian rate.
The Australian War Memorial memorial website states that Australia's suffered one of the highest casualty rates of the war in regards to total men embarked, that is total number if Australian troops to enlist and leave Australia. The statistic they state is that 65% of the men embarked become casualties in one form or another.
In terms of why this figure is so high, The Australian troops were generally very good soldiers, especially in the latter half of the war when there were large numbers of experienced veterans. They were also known to be very aggressive in battle and sometimes unconventional making them a suitable choice for use as shock troops and were often deployed at the forefront of large attacks. The Canadians were also used in a similar role for similar reasons. Being at the forefront of an assault naturally means sustaining large numbers of casualties. By mid-1918, reinforcements from Australia had just about dried up and the high casualty rates meant that units were operating below full strength. To address this, some understrength units were disbanded and used to bring other units up to strength. There is a great story of these units rebelling against their officers in protest to this policy. In the end, a compromise was made. Those units designated to be disbanded were deployed in the front wave at Amiens in August of 1918. If they survived, the men agreed to disband.
There is a certain amount of criticism levelled at the commander of the Australian Corps, Lieutenant General John Monash for his decision to continue pushing the under strength Australian divisions throughout the Hundred Days campaign. In particular, his remarks to one of his divisional commanders just before the Battle if Mont St Quentin, "casualties no longer matter," have drawn significant criticism.
Sources: Australian War Memorial website
The Great War by Les Carlyon
The Official History of Australia in the War of 1914-1918 by C. E. W. Bean
The Australian Victories in France in 1918 by Sir John Monah
Monash as Military Commander by Peter Pedersen
Edit: military casualties per capita.