It makes the replayability better - have backup sets, have to grind new backup sets. In the event of no backup sets, bave to grind all new gear. It's refreshing to actually see upgrades and also to plan for the inevitable glorious death/DC of the player. It adds that to the game - you never truly reach a point where you have it all and you're bored from it. If you ever feel that way, you can always go balls deep in a higher GR and either emerge victorious and repeat the process, or die valiantly. The game never stagnages in terms of gear progression!
The thrill - getting your cheat death procced in a higher GR against a RG - nothing in softcore can prepare you for the rush and panic of suddenly not knowing how to act/react when you are fully aware of the fact that saving a cooldown for later might get you killed right now, or blowing a cooldown right now might get you killed in 10 seconds. That's the rush, when cheat death procs and you're staring right into the abyss. That feeling is the embodiment of HC.
Camraderie - If you play with friends, you can act like jackasses in trivial content, but when you're in deep shit, the level of coordination and teamwork and situational awareness of not only yourself but your teammates - this kind of 'bonding' doesn't happen in softcore. That moment when you DC but everyone is orchestrating minion sniping and health globe diving, that builds more trust and lasting team friendships than anything on SC. Or the occasional valiant death trying to save a DC and the subsequent powerleveling and gear grind. There's a code of honor, of sorts, in multiplayer HC. It's hard to explain but if you play HC long enough you'll eventually see that DH that always hangs out quietly in the Vault in to the melee and start shouting orders and you'll suddenly think very highly of him. And you'll feel indebted to him for being around him. You never really get that in softcore. It's just hard to describe, but you learn a lot more about teamwork and you learn a lot more about your teammates through HC.
The accomplishment - getting through something semi-challenging feels much more rewarding knowing you pushed yourself from level 1 to level 70, from yellows to set pieces, from Hard to GR40+, and you just survived Hamelin's octagon ring of death and lived to tell the tale. Another thing that might not feel so rewarding on SC.
It has its drawbacks though, it's not for everyone, but if you're bored of SC it's very much worth a try... or two... or three!