The puff-n-stuff look is a bit of a mystery it seems. While the reason poodles are able to have these haircuts are pretty obvious, because they are one of a few dog breeds that have hair (that keeps growing) rather than fur (that sheds at a certain length), they're not the only one! Likewise, the oldest images of poodles show some more interesting haircuts but nothing standardized or very extreme like the modern show clip.
I have read two competing theories on various parts of the Internet - one, it comes from their sporting heritage and it is to keep joints warm while minimizing amount of overall coat for burr collection; or two, it comes from their pretty long pedigree as circus dogs. So what's the real scoop? When and where did these haircuts come from?
My girlfriends mother is a big dog breeding enthusiast and I also found some sources on the internet to back this up:
Poodles used to be used to retrieve stuff in the water ("(The word "poodle" is derived from the German pudel, short for pudelhund, which means "water dog." Pudeln in German means "splash," and is also the root of the English word "puddle.") " So the reason you cut them the way you see today was for practical uses. An excess amount of fur would weigh them down in the water but you still wanted fur around some vital places (around organs and joints) to keep them from getting too cold from cold water.