The Smash Bros. series follows the imagination of a 9-year-old boy and ages with that boy in real time.
In the first game, the child is 9 and imagines these toy statues to be fighters that he smashes into each other until his parents, represented by Master Hand, tell him to clean up his toys.
In Melee, which came out 2 years later, the child is 11 and his imagination has developed to make the playtime more complex, but things still go basically the same way, them wanting to play against their parents' wishes to clean up.
Crazy Hand is the representation of the child's own playful and spontaneous nature but also of chaos while Master Hand is a symbol of order. Brawl is a 2008 game, and takes place when the child's 18. In that one, he's less concerned with what his parents tell him about structuring his playtime and more worried that he can't take his beloved toys with him when he goes off to college.
He thinks that the other young adults wouldn't approve of him having a bunch of kids' playthings in his room, and thus in the alternate dimension where his fears are enacted out in a realm where his toys have lives of their own, the villain is literally just a respelled "Taboo", as in, "others won't let me do this" and the statuefied fighters are representative of the child's wanting to tell those around him that they're just collectibles without the personal meaning he's given to each of them.
4 is in 2014, which would put the kid at age 24. It doesn't have a world-threatening storyline because in the previous game the kid figured out that he doesn't need to care about what other people think and at this point he has his life pretty together.
In Ultimate, he is having a kid of his own and is internally struggling with his desires to abandon his wife and keep playing with toys forever and his sense of duty to stop playing with toys and focus all his energy on creating a safe and stable home for the child.
The ending of the game is that the universe collapses completely if a perfect balance isn't struck, and when the balance is found, he realizes that he can use his imagination to play with his kids, passing down the "spirits" to them as well.
Oh, and it's never specified that the hand that chooses characters and the hand that animates the characters in the intro is specifically Master Hand. So that could be the kid doing that as what will eventually become Crazy Hand.