They're not usually that fluffy. Typically the owner shears them before the hair gets too long, but before shows they sometimes keep the coat on a month after they normally shear it, then tease it out to give it the extra fluffy appearance like in the picture. Once the show's over they give them a haircut. Even when they're all permed out they take special care to leave the face and...er...genital areas clean to ensure that the critter is still comfortable. Plus it's not like they just leave them in their cages when they get like that. Angoras require A LOT of care to stay happy and healthy, as in being waited on hand and foot every day, and owners have a big incentive to do so even if they're only raising them for their fiber because a happy, healthy animal gives the highest quality yield. You can tell from how white the bunny's hair is in this picture that the lady probably has been grooming and attending to this little creature constantly. Not to say there are never abuses, but they're usually done in a factory farming context rather than with certified owners like Ms. Chu.
Another fun fact: if you know how to do it correctly, you can spin wool right off the rabbit. Certain breeds of Angoras molt their coats every few months, and when they do the coat comes loose and can be gently pulled off without hurting the rabbit. I had a very fluffy dog who shed like that; the patches of fur that were being shed off would become discolored and stick out, and it was oddly therapeutic pulling off the shedding chunks. You can see an example here: http://www.youtube.com/watch?v=IY5ccnuC2yE