I think the main reasons have already been mentioned, for me it was really all about the novels. We did start some Dragon lance campaigns including the first module of the official Dragonlance Chronicles adventure but I think we got stuck between do you act out the offical story/do your own thing conundrum that I've heard people talk about with licensed RPGs.
We all loved the books though and as far as I go the main things that stick with me 30 years later are
Chronicles and legends
(Where it all began for me and my friends. I reread them both not that many years ago and I felt they both held up well. Better than the Icewind Dale trilogy by a long shot. At the time I preferred Chronicles - I guess the classic hero's quest and clear evil to be defeated made it easier to like than legends many greys and shadows and not entirely being sure who you should be cheering on but rereading them I think legends depth took the lead. Its like when you watched Return of the Jedi as a kid and you loved the Ewok bits on Endor but Luke and Vader on the Death Star was too talky talky but rewatching them when older you realise that throne room was where all the intresting character bits are and the Ewoks are just stupid, except Chronicles is still good and not the galactic empire's finest troops being defeated by a teddy bear throwing a stone at them.
People on the good side who were jerks (looking at you leaders of the knights of Solamnia) or at least flawed (keep it in your pants Tannis).
Bad guys you liked but were still actually bad. (Kitiara, Lord Soth)
Bad guys you liked but weren't sure if they were really bad guys. Or if they were, how bad they were (Raistlin, Dalamar).
Huma and Kaz the Minataur
The art. Oh yes the art!
Pictures of Flying Citadels.
DRAGON LANCES! I mean people riding dragons with magic lances engaging in dog fights with other dragons. Seriously how cool is that!
Kenders, gully dwarves and gnomes (yes all of them - especially the gnomes)
Characters turning out to be a, gods b, dragons c, dragon gods d unexploded sentient spells.
Heroic angsty deaths, tragic deaths, tragic backstories. Emotional catnip. (you'll buy 20-30 more books in the series long after the quality has dropped to keep getting those hits)
The Cursed grove of the high tower of sorcery in Palanthas.
Epics wars and personal stories wrapped up in each other.
God's that are actually interesting.
Weasle's luck and Galen Beknighted - two stories about a shifty son of a podunk knight of Solamnia in a swamp based castle facing a small local evil. A million miles away from world level battles of gods yet still Dragonlance.
Actually both the Heroes and Heroes II trilogies. Storm bringer and Gates of Thorbardin were good too. After these books only really the preludes held up the quality by Preludes II it was going downhill fast.
One last thing my friend used to joke that Weekender by Flowered Up was actually called Wee Kender! and we ended up with a running joke that if Kender came up in a conversation we would start singing it.
Just to note Weekender is a) one of the greatest musical achievements of mankind and b) a 13 minute long structureless track that got the band dropped from their lable that embodies the Baggy/Madchester scene (although they were from London) and appeals to only approximately 0.000003% of the worlds population so your mileage may vary.
But my nostalgia runneth over so it's too late to stop now.
So one last time. All together now.
"Wee Kender! Wee Kender!
Go out - Have a good good, good good time"
Edit - just remembered there is a moment of swearing about half way through so NSFW.
Even if I repeat some points, I'll post why I absolutely love DL:
Sense of discovery/exploration is ingrained in the setting: the gods, the dragons, pre-cataclysmic ruins like Xak-Tsaroth, isolationist civilizations like Silvanesti.
Legacy: we could say recovery instead of discovery, as a big part of the adventurer's role is restoring a lost, past and they strive to become continuators of this better past as they uncover and own storied magic items (Dragonlaces, the Orbs, the Disks of Mishakal) or become members of legendary organizations (Knight of Solamnia, Wizards of HS).
The shock and contrast as this mythic, high-magic past suddendly clashes into a gritty post-apocaliptic world. I particulary liked the sorry state of the fragmented and xenophobic polities of Krynn as it is both realistic and provides the PCs with yet another task - to mend the bridges between human, dwarf, elf, gnome...
High charisma villains that you actually empathize with: Raistlin, Kitiara, Dalamar, Soth. Even 2nd stringers like Verminaard and his Nightbringer mace are memorable.
From a player perspective, I always liked the influence of the 3 moons on wizards. I’m not sure I’d say I liked that most, but I did like it and it’s a bit different from most of the other perspectives here.
I liked the novels when I read them as a kid. The grand scale of the novels was epic. Raistlin was an interesting character and it was cool seeing a character that didn't fit the traditional heroic mould, someone that didn't have a good alignment but was still a part of a good adventuring part.
I quiet liked kender in the novels, they work in literature, but their kleptomania could be problematic in game. I like the direction they took in the UA release that made them still seem like Thieves without being a problem to the party.
But I think the thing I love most about the setting is the Draconians and their dangerous death throes. I always thought they were cool and dangerous, especially when you killed them. I always liked Draconic things and these evil looking dragon-men were scary when they trapped your weapons in their stone bodies or other effects when the died.
Most of what has already been said. The Chronicles and Legends were formative for me, and helped define what I want from a fantasy story. The sense of history in the world, and in particular how that history was directly used and explored, was a significant part of my choice to get my BA in history.
A few other things that stand out: Lord Soth, draconians, Raistlin, Wizards of High Sorcery in general (especially the Towers), the Knights of Solamnia and their opposite numbers the Knights of Takhisis, and tinker gnomes (can't help it. Wacky inventors are cool).
I love Dragonlance. I got the first novel when I was in 5th grade and it was formative for me. Still heavily influences my fantasy worldbuilding (of which I think it's an excellent example). As for the particulars, I'll just echo what others here have said.