As with art, the "beauty" of chess moves lie in the eye of the beholder. Sometimes sequences of moves are interpreted as "beautiful" perhaps because the player themselves do not posses that calculation ability and are in awe of that.
However, I don't think this only has to do with calculation alone. Some patterns or checkmates look "beautiful" as well. I think when something in chess is called "beautiful" it generally either means:
a) The one saying it subconsciously acknowledging they didn't see themselves finding that solution if it was their own game
b) The result was instructional and typically of simplicity within some complexity. This one is tougher to describe, but a smothered checkmate is a good example of this because the calculation alone is not that difficult. In that case, it is instructional (and rare) how a lone Knight can deliver checkmate, yet one piece giving checkmate has the curtain of simplicity.
Notice that most "beautiful" moves are rare (or less common if the motif is known). Castling checkmate, en passant checkmate, smothered checkmate, B + N + K vs K endgame, or basically any attack in a romantic era chess game could all be considered "beautiful."
I've been playing chess for 2 years now and I still have no clue, do they just call moves that require more calculation "beautiful"? This has been around my mind for quite a time now