Much Australian slang is derived by simple abbreviation of longer words. Thus, television becomes "tellie", barbecue becomes "barbie" and Australian becomes "Aussie". Note that down here, "Australia" is pronounced "Oz-trail-yah", so "Aussie" is pronounced "Ozzie".
Now, "Ozzies" have to come from somewhere; logically, they come from Oz. There is also, of course, the literary reference to the Land of Oz of L. Frank Baum's fantasy books - a land so far away, it's impossible to go there by ordinary means, full of weird people and bizarre creatures.
As an aside, you'll also note that we believe there are only three syllables in "Australia", rather than the four that everyone else seems to want to use (most North Americans I've met tend to say "Ore-stray-lee-aah"). So here's a handy tip for Americans wanting to pronounce "Australia" like a local: say this Spanish word:
Estrella
Apparently, many Americans know enough about proper Spanish pronunciation to know that words ending in "-lla" have an additional "y" thrown in before the "a", giving "Es-trell-ya". Slide the "s" slightly more towards a "z" sound, and viola, you're saying "Australia" in a near-perfect Aussie accent.
