The terminology in FF13 might not make much sense at first, but after several hours the words Fal'cie and L'cie should make sense if you've paid attention.
The relationship between Fal'cie and L'cie is an interesting one. Lightning describes L'cie as "pets" of the Fal'cie, but as you learn later, it's not quite that simple. But it's mostly true, and the Fal'cie have godlike power over humans in this universe. There's an unsettling sense of power that the Fal'cie have, that (at least for me) encourages one to reflect on the power that we, as humans, wield over other creatures. From the perspective of some L'cie in FF13, you feel the weight of that power as you explore the suffering that it inflicts on the human inhabitants of this universe. I think FF13 explores this relationship thoughtfully without getting preachy.
All worthwhile stories, in my opinion, say something about the human condition. Up until maybe 50-70% through the game, I wasn't sure this game had something important to say about what it means to be human. But I think once you learn a bit about Vanille's and Fang's history (i.e. their previous act of genocide) and their willful amnesia, the actual story starts becoming clearer.
Rejecting genocide in the face of torture/violence, atonement for past genocide, fighting for the good of others regardless of the inevitability of failure and its consequences. Thinking about it, these remind me a bit of FF6's underlying themes, which is maybe one of the reasons I love FF6's story so much as well.
As for a summary of my opinions, I think the story and ideology of FF13 is definitely one of the most mature in the RPG genre. The art is beautiful, and I adore the soundtrack (I think the FF13-2 soundtrack is even better). I thought the dialogue was pretty decent for a non-Mother game, and the English voice acting was solid. The characters were some of my favorite from any FF game, especially Lightning. Gameplay was refreshing and fast-paced. I agree the upgrade system could have been improved. Definitely my favorite Final Fantasy.