Non-serious reply: because it's the best FF, lol
Serious reply: What makes FF8's story is so great is how every thing is tied together, even the small things. In FF8, unlike most FF's, everything isn't shoved in your face. A lot of key story points are left to the player's imagination, such as Rinoa being Julia's daughter, or Squall being Laguna/Raine's son. The movie Laguna was in on disc 3 influenced Seifer so much it pretty much made him who he was. Notice how when Laguna uses the gunblade against the dragon, he fights with Seifer's stance. Where do you think Seifer got the stance from? At the beginning of the game they say a movie called The Sorceresses Knight is on the waiting list, most obviously Seifer wants to watch it again.
Another facet that makes FF8's story so appealing, to me, at least is that it's a time travel story. I love time travel stories. The entire game is one endless time loop. If Squall hadn't gone back in the past along with Ultimecia in the ending, then Ultimecia wouldn't have given her powers to Edea. But at the same, that very moment Squall gives Edea the idea of Seed and Garden. If this event didn't happen, Garden and Seed wouldn't exist, but at the same time Edea wouldn't have been possessed and who knows what would have happened.
Knowing this, Cid and Edea already know Squall's future because the future Squall already told Edea before she could become possessed. So when Cid pushes Squall in the game, it's not only because he's a natural leader, it's also because he knows that Squall's the one that defeats Ultimecia in the first place, so he pushes him to do this.
Ultimecia is defeated, but the heroes of FF8 keep doing this over and over in one endless time loop. No one know which loop number it's on when the game starts.
Thematically, the game has a lot more themes going for it than most other Final Fantasy's, especially since they all aren't shoved down your throat, like in VII or especially IV. The themes of destiny, considering the different philosophies of enemies, and defeating one's inner demons were never been pulled off so well in a Final Fantasy.
Story wise, what Final Fantasy bar VI has better story pacing and memorable moments? The Dollet mission, the Laguna flashbacks, the parade and attempted assassination on Edea, the Garden war, the missle base and attempting to stop these said missles hitting Balamb Garden, Garden flying in the fucking air, Esthar, going into space, the Ragnarok, and more? The entire game has memorable moment after moment and not one of these moments is dull. The story is action packed and never lets up.