For me specifically? Well...
FFVII introduced me to a lot of things new to gaming at the time that revolutionized it. FMVs for one, probably the best quality in existence at the time, and incorporating several of them into the scenery instead of having them viewed as a separate cutscene. Yes, Cloud's lego-tastic body doesn't really mesh with the beautiful backdrops, but just the fact that you could run around in the middle of an FMV in progress (better displayed later in the next game where there was a war scene where there were both polygon soldiers and FMV ones fighting with each other that you could also run into and fight, or choose not to). Also the first PS1 game I ever saw, making a HUGE first impression visually with summons, spells and the like.
But enough about dem gray-fics, which is something I cared about much more fifteen years ago. Having played FFVI before this it was a massive shift not only in setting but tone from fantasy to cyberpunk. Like a common compliment I see paid by fans of Sonic SatAM, Shinra Inc. isn't seeking to take over the world- they already have. Shinra runs almost everything, and the only group opposing them gets massacred less than 2 hours in. Only after leaving Midgar do we get any indication that every town isn't like that (or that massive, and being younger I could actually imagine a game that vast happening). Last game, we had 'Son of a Submariner!', and this time people are swearing like sailors. To me it was like someone going from Adam West Batman to Frank Miller Batman.
But Shinra's not even the worst enemy, oh no. There's also a spree killer with the world's longest katana on the loose trying to outdo Jason Voorhees. When I first saw this game I witnessed a friend play it up to the Kalm flashback sequence. Another first- never before had I seen a game go into this much detail for a villain's backstory before, nor make you a direct participant, with the highlight probably being the moment when Sephiroth's theme begins for the first time, signifying his sanity officially snapping and punctuated by one of the game's best FMVs. It would later be used to signify that you're about to travel through another Shinra facility where he's painted the walls and left behind monsters for you to deal with instead of soldiers.
Later on I would play the game myself and appreciate other things, like the Materia system (which I still believe to be the best progression system in the series), or the deconstruction of Cloud's character. In short, it came out at the absolute perfect time for my age and made the perfect first impression, as I figure it did a lot of people. Things have changed a LOT since then and I probably won't get the remake, but when it came out there was simply no other RPG that did the things that it did, and I'm speaking here as a lifetime fan of Chrono Trigger and FFVI.