Without a bit of time to think I can't hazard a guess, except that it is taught to all schoolkids here, and has somehow developed the credentials, just through chronic repetition, as the hallmark song for Oz. It has been used at a lot of sporting events, and other events, as a sort of unofficial anthem. If you play Waltzing Matilda for an Aussie who is overseas you'll probably make them break out into a grin.
And regarding anthems, we haven't ever really had a good Aussie anthem. We, well the true-blue Liberal party supporters, always clung to the Monarchist anthem, God Save the Queen, but then we decided to get another anthem which was more specifically Australian, but instead of choosing a song which a lot of people liked, and which most people could sing, the powers-that-be chose Advance Australia Fair, which is pretentious, trite, very twee, and very difficult for most people to sing. There doesn't seem to be a key which allows all sorts of people to sing it comfortably, and the words are amazingly unrememberable. If you ever get a chance to watch a bunch of Aussies having to sing through it you'll be able to count on one hand the ones who actually get past the first verse, and half of the crowd won't even make it through the first verse. Part of the Oz tradition is to value sincerity, or to be "fair dinkum" and those lyrics are not fair dinkum, in my opinion, so it makes it very difficult to sing that song at all, let alone with feeling.
Add to all that the links within Waltzing Matilda to our history, and you have another reason why a lot of average Aussies prefer to sing it than any jumped-up attempts at squishy sentimentalism. The attitude which Mr & Ms Average tend to place value on are not about sheep stealing (theft is actually opposite to the inherent good old Aussie values) but about defying authority (re: our convict heritage, and a lot of Irish versus English political conflict), and self reliance in adversity, (the swagman making his own way in the world and relying on his skills including basic survival skills in a harsh environment), and the mischievous humour of the situation, and just giving the politicians the finger by wanting to play a totally politically incorrect song instead of the snivelling drivel that they want us to willingly submit to.
All right, so I didn't need that much time to think about it. :-)
But these are only my opinions, and my conjectures. Other Oz 'Catters will undoubtably pitch in and give us their reasoning.