I don't think I've ever seen a fat sloth like I've seen a fat cat, dog or human, and for the most part this is what they call exercise. I know they tend to eat proportionately to their activity level, but there's gotta be some who pig out more and are lazier than others just like anything else.
I would imagine that sloths tend to be cared for by people who have their best interests in mind (zoos, animal sanctuaries, etc), and therefore are fed a reasonably healthy diet. If you could buy a sloth at the local pet shop, then there would be tons of videos on youtube of fat sloths in baby clothes owned by fat people who don't know any better.
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