I know what you
Americans are thinking:
"Oh you wacky Aussies. Only you could come up with a silly name for something sensible."
Another wonderful piece of Australian English vernacular, this one is a little harder to pin down, origin-wise, hence no reference to origin, or Usenet first-use. (Wertperch and others have commented that this same term is also used in England.)
In Australia, we don't use the, well, quite silly, word "fanny", even though it is used in Mother England. BlackPawn reminded me that in Australia, fanny is used as a deprecated and derogatory word for vagina - but mostly amongst immature school children. Americans use the word to refer to the derriere, the rump, the bottom. In America, what we call a bum bag, you call a fanny pack.
In Australia, we pull no punches. We call a spade, a spade. It's a bum, people! You know it, but the right wing, bible belt censorship laws in the good ole U. S. of A. mean that you call all sorts of things by less... offensive names.
So. It's a bum. There also exists a need to carry small amounts of stuff with you at various times in your life. A camera. Your wallet. A train ticket or maybe even a sandwich. Yet at the same time, you want to keep your hands free to do other, more important stuff.
Luckily, we homo sapiens are designed quite well in this respect. We have hips, and just (hopefully) below, at the back, a mass of gluteus maximus called the aforementioned BUM. with a simple band around the waist, and a BAG attached to this, you can rest said bag on said bum.
The bum bag.