In the first Super Smash Bros, the characters were dolls of some sort. In Super Smash Bros. Melee, the characters were trophies. These are witnessed by their respective Adventure ending scenes when the characters regress to their, um, original forms (for lack of a better term).
I'm not quite sure whether or not the Hands were purely behind the transformations since the story isn't exactly elaborated on other than a few short cutscenes.
EDIT:
Originally Posted by Crazy Hand Trophy Description
Where the Master Hand loves to create, its alter ego is impulsive and destructive, consumed with that hollow feeling which comes from destroying one’s own creations.
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Upon further inspection, the Crazy Hand trophy description does say that the Master Hand loves to create, so it is possible that the Master Hand created the Smash Bros. universe.
Jam it back in, in the dark.