I'm back again, it's almost like I think more than 3 people are going to read the long winding critique I'm posting here, but I got offtrack in the first post talking about the story, so I figured I should at least mention the gameplay, and go into why I think the game gets so much fervent hate.
Let's face it, turn-based fighting featured in most JRPGs isn't exactly the most exciting gameplay in the world, even fans will generally tell you that JRPGs are mostly about story, atmosphere, and very few will sing the praises of the old school turn based combat model. Even then, there are exceptions, but I don't think very many people will count FF XIII among those exceptions.
FF XIII has the paradigm shift system, and basically takes away control of your other party members entirely except to switch their roles mid-battle, For the first half of the game, anywhere from 13-20 hours, you can basically get away with minimal paradigm shifting, probably because the character split means you are often stuck with only two characters, and only a handful of roles to shift between before you unlock your full party. Almost all of the complexity and interesting aspects to the system are basically non-existent until you get about 20 hours into the game.
The leveling up is facilitated by the chrystarium, basically a poor mans sphere grid with even less choices, during that same 20 hours that severely handicaps the paradigm shift system, the chrystarium gets equally handicapped. At the beginning, each character only has a few paradigms, and there are hard-locked walls that you can't get past until you get to certain points in the story. What this means is that the first 20 hours of gameplay, any sort of grinding is pointless, you are locked into specific roles, and you can't even level those roles up beyond a certain point, the game basically level caps you until you beat the next boss. This wouldn't be so bad if the combat system was wide open and encouraged experimentation or playing around, but it isn't, you don't even get introduced to the concept of staggering enemies until about 5 hours into the game, and that is probably the most active part of combat until you get your full party and actually have the room to play around with paradigm configurations.
The chrystarium really doesn't serve any purpose except looking pretty, what few branches there are in the chain only branch off by one or two points then you go right back to the chosen paradigms main upgrade path. Once the chrystarium and paradigms open up in the late game, its already far too late to really generalize your characters as they'll already have a crapton of levels in the roles they were railroaded into at the beginning, so unless you really love the combat and want to grind out the other paradigms, there's little reason to bother switching leveling paths until the very end of the game when a characters main paradigms are all maxed out.
The magic system is also kind of silly as the main story feature of being a l'cie is the ability to use magic, but the magic in the game is limited to tossing tiny balls of elements at the enemies, it would be fine except they decided to make the magic thing a story point, so it looks a little silly that the organized military with jetpack snipers and giant fucking death robots is apparently afraid of something that looks so visually unimpressive in combat.
Since this topic is only about XIII, I'm only going to mention the sequels as being kind of seen as more of the same except the story makes even less sense (time travel basically makes the entire story of the first game pointless, and all the characters are now completely different, trying to skip from XIII to XIII-3 will make your head explode), XIII-2 apparently cut out the finale to the game as DLC, and the trilogy keeps pushing lightning as the protagonist, which despite her popularity in Japan, the West is mostly indifferent to her, and are largely sick of her at this point. These factors have probably kept the hate train for XIII alive longer than it would be normally, because a lot of Final Fantasy fans who disliked XIII just want it to die already so Square can give them a new story, rather than keep trying to push the story and world of XIII on us.
As for why the game gets so much vehement hate? A lot of reasons really, but a lot of it is that the game is just boring to a lot of people. The combat plays itself even more than most RPGS, and even when you are actively involved, you're mostly just choosing roles to adapt to enemy attack patterns, once the pattern is down the battles become a chore, and most of the strategy of the system is removed, you feel less like your playing as these characters, and more like a pokemon trainer who just throws suggestions as to what you want the heroes of the story to do. A lot of people just couldn't make it through the 10-20 hour slog to get the game to open up a little bit, and even after opening up, a lot of people just didn't care about the characters, story, or world anymore.
The game wasn't broken or actively bad like some games, but many people saw it as boring, and a chore to play. In some ways, that's actually worse than being broken or offensively bad, at least in those games, you know what you're getting, and can actively mock it. With XIII, I think some people got really worked up about seeing a once lauded series known for its fantastic worlds, and epic battles and stories, reduced to 20 hours of tutorials and corridors, that doesn't even attempt to flesh out the world beyond what they can show in pretty cutscenes. Empty spectacle, ultimately accomplishing nothing; that was my experience with the game once everything is said and done.