Well, the origins of the various armors have been pretty well explained by people so far, and the reasons of the design of DRG too. In all honesty, more than just DRG are the way they are due to appearances in previous titles (ex. WHM is obviously another iteration of the traditional WHM garb), but I like the aerodynamic flying weapon explanation for DRG. It's a little more lore friendly. (This got long, excuse the spoiler tag abuse.)
PLD
Like people said, it's the official armor of the Sultansworn. Likely the white was chosen to represent purity and fidelity (likely the same reason it's been used in previous FF games and on Paladins in other titles). While gold trim is also not breaking any PLD molds, you could also say it's tied to Ul'dah's wealth without making too much of a stretch. The armor underneath is relatively nondescript plate, so is probably just for protection and nothing more.
WAR
This wasn't exactly passed down from Curious Gorge, no, rather he helps you find it. The armor he wears is just a replica of yours, which is the real deal (the armor of his ancestor, who was a true Warrior). If I remember correctly, words of power are supposed to be carved into the armor, but they're obviously not visible in the design. I decided to justify that they're probably on the inside. The Way of the Warrior was developed and taught in a small Hellsguard village removed from any large city, so all the leathers and furs were likely just the materials that were readily available (with metal being used sparingly, since there wasn't a lot of it to go around). The village was also far North, so likely cold, which would again explain the use of furs (to add some more warmth to the protection of the leather). I don't have a good reason for that bladed wrist guard they seem to wear, but I'd guess it's simply another weapon in case the Warrior was disarmed or if an enemy was too close to get a proper swing at them. Also no good guess for the horns, but the original Warriors did lend themselves out as mercenaries, so there was probably a benefit to just plain old looking intimidating. What better way to do that than welding giant horns onto your helmet, right?
MNK
No idea. As far as where it comes from... the AF is worn by all the Monks in the Fist of Rhalgr and is imbued with special powers. It attunes to the wearer, it strengthens their chakra, and the cloth itself has been imbued with Rhalgr's power so that "only monks of his Fist may don it" (though I suspect that part might be myth, the Twelve don't seem to intervene with things so directly, but who knows). As for the color or that giant feather in the circlet, I have no guesses at all.
BRD
ZephyrFox already explained how it is the player comes to have the BRD AF, so I won't rehash that. Before Jehantel hid them all over the place they were given to him by Pukno Poki. So as for the design... I don't know. Moogle fashion?
BLM
I don't believe it's ever actually explained where BLM garb came from in the first place. We're told that in order to close the Voidgate we'll need it, and as Laevetine mentioned, it does look exactly like Ququruka's, minus the color. I don't remember it ever being said that it actually was his, though. As for the design, I'm gonna go with non-Eorzean staple again. The hat is pretty witchy and the style's been used by BLMs since at least FFIV. The color is hardly an imaginative choice for a black mage. No guesses for the fancy purple bits, but I guess sometimes embellishments are just embellishments. This might be one of those times.
WHM
The origins of this one already got talked about, and like I mentioned at the beginning the design's pretty traditional Final Fantasy WHM gear. Like with PLD I'm going for white = purity. I don't have any lore-friendly things to add beyond that.

Originally Posted by
Laevetine
WHM made little sense to me seeing as you collect the rest of the gear yet the body piece is exclusive to you and you alone and did not exist up until that point.
I can clarify this, though. The
design for the body piece does exist before you get yours. The gear was worn by the ancient White Mages (probably the ones that brought about the Water Umbral Era... that's a great way to stave off an angry elemental of wrath, right?). Oha-Sok becomes your body piece to keep an eye on you, and simply takes the form of the final artifact of succor which you were unable to find. Heck, for all we know the mages of old may each have had a special bond with a single elemental who then formed their body piece. It'd explain why we couldn't find one, at least.