On a superficial level, any four-syllable word that flows well sounds beautiful to the ear.
See: Diarrhea. It's a beautiful word superficially but the contents leave much to be desired.
But there's more:
When I pronounce "Shenanigans" out loud I hear "She Nags Again"
Nagging (again) is sadly is one of the top complaints husbands have about their wives. It's a cause of divorce. What can be beautiful about that?
Quite a bit. By turning "She Nags Again" into a funny word like "Shenanigans" you're making light of it. You're turning it from something harsh to something that's not a big deal. "I might be nagged at but I'm not taking that nagging personally."
Making light of otherwise dour situations, not taking things personally, putting poetry to language.
That's shenanigans and that's beautiful.
