Final Fantasy I/WMG

The Four Fiends are the Four Light Warriors from the past
After committing attrocious acts against mankind they were cursed to become horrible monsters. Chaos, who is God, then made them relive their deaths at the hands of themselves in human form and created the time loop so they would die over and over until the end of all existance as punishment.

Chaos threw the battle at the end
Kind of lame, but this place seemed empty. Basically, if everything really was a stable time loop then this game should be exactly like last repetitions with only Chaos being aware of the changes.

After a near infinite repetitions of a Groundhog Day Loop lasting thousands of years a go, so Garland/Chaos just gave up on trying to come out on top. He could win, but he failed over and over in maintaining his sanity long enough to do anything, but be beaten and sent back in time to do it all over again.
 * That sounds an awful lot like Shade Impulse, actually.

Due to a Stable Time Loop, Chaos either never actually died or the Light Warrior lost their Powers after beating him.
Killing Chaos means the Four Fiends can't exist in the future. That means there's no reason for Chaos to exist in the pact. That means the Light Warriors aren't needed anymore, therefore in the next iteration they may very well not exist. ALTERNATIVELY, killing Chaos would cause a Grandfather's Paradox since there's no way for the Light Warriors to beat Chaos in the first place if Chaos was already dead. The moment things go back to normal and that "Garland" is fine and dandy, he decides to go kidnap Princess Sarah all over again...and dying, which would somehow start the Cycle. Final Fantasy Dissidia mildly supports this theory. Maybe. I don't know actually.

Being defeated actually made Chaos a true God.
By killing Chaos and ending the Time Loop, it also caused a massive Time Paradox that would threaten the stability of the Universe. In order to make sense of this, the Universe itself elevated Chaos to Godhood to prevent total time distortion. This may or may not have been his actual Plan, but either way Chaos has become an Actual God in charge of Discord. He then spends the rest of his time playing Checkers with himself in the Past. And occasionally taking a temporary Avatar like in FFXII.

The four fiends are really the Warhammer 40,000 Chaos Gods trying to edge their way into conquering another universe
I mean, they already did it with Warhammer Fantasy, so how hard could it be? Here's the breakdown of who's who:
 * The Lich is Tzeentch, ageless and wielding powerful magics. He also came up with the idea of creating the Stable Time Loop.
 * The Kracken is Nurgle, because a it's big and disgusting looking Eldritch Abomination.
 * Being the only female fiend, Kary Marillith is Slaanesh (she's rather... seductive too)
 * Not to burst your bubble, but Tiamat's a chick, too.
 * Which leaves Khorn to be Tiamat. I mean, come on, Khorn is the god of ruthless bloodshed, and Tiamat is a giant three-headed dragon! How much more ruthless can you get?
 * War Mech is Malaal... just because...
 * And as a consolation prize, aspects of Gork and Mork are within every random encounter.
 * And they wanted Garland to be the next Horus, turning him into "Chaos" in order to forge a permanent link into the FFI world.
 * Since Garland threw that last fight, the Gods would try again with another universe, leading to Final Fantasy II

Culex is the fifth Fiend.
He was Genre Savvy enough to have known Chaos would lose, stole power from the other four Fiends (the crystals that aid him in SMRPG), and escaped.
 * Then why does Culex play Final Fantasy IV music?
 * Further, the original release of Final Fantasy I had elemental orbs, not crystals. Only the rereleases changed it, and Super Mario RPG came out long before those.
 * Actually, they were always crystals in Japan. They were changed to orbs for the original American release because of space constraints.

Garland is the fifth light warrior
He has a dark crystal in his palace, probably representing time or darkness or both or something. He turned to serving evil however.

Chronodia started the time loop
Chronodia is a bonus boss in the 20th anniversary edition, resembling a sphere that has absorbed the fiends. Its dungeon is connected to the Temple of Chaos. That's pretty significant circumstantial evidence.
 * Alternatively, to answer an issue from the Headscratchers page, Chronodia is the entity that existed in Chaos's place in the timeline before Garland took over. Chronodia continues to exist, but has been shifted out of its role as god of chaos.

Garland went along with the Four Fiends because he was angry at the Light Warriors.
This is why he continues the time loop, despite the fact that the Light Warriors will just kill the Four Fiends in the future. He doesn't care about their plans, he just wants to make the Light Warriors suffer.

Final Fantasy I is our world, after the return of magic accelerated continental drift.
Onrac's continent is so North America, it's not funny.

Final Fantasy V is a prequel.
Which explains why you see Gilgamesh in the Dawn of Souls remake.

White Mage is, and always has been, a dude.
Mostly because it'd irritate the fans who insist the heroes used to be drawn much manlier. They've always been effeminate!

Soul of Chaos is the conclusion to the story in Dissidia.
Dissidia ends with the biggest masterminds of the Endless Cycle still at large- Cid Lufaine and Shinryu. Over the course of the games we find out that Shinryu, Cid, Chaos and Cosmos are able to draw beings from other worlds to the ones they inhabit. We also discover, in scenario 000, that Shinryu is able to create dream-worlds to torture his victims.

FF 1 sees peace given to Garland, who is also Chaos. Chaos' previous identity as Cid Lufaine's son, however, is not yet destroyed, and serves as the titular foundation of the Soul of Chaos. So C is a dream-world created by Shinryu to torture Chaos, as well as to feed on Chaos' power and hide from Omega and the Light Warriors. This is why So C contains world maps and dungeons and beings from other worlds in the FF series; as these are places Shinryu and Chaos have been or influenced.

As of 000, Cid Lufaine has again relinquished his moogle form, and is free to roam the multiverse, while Cosmos rebuilds World B. Cid decides to put down Shinryu before he can find another man to start a new Endless Cycle with. So Cid pulls his second son, the Warrior of Light, and his companions, who have been deposited outside Corneria for the second time, having undone their journey for all but themselves, breaking Chaos' paradox, into the Soul of Chaos to find and destroy Shinryu. It is here that the warriors put an end to Shinryu once and for all, or at the least punish him for his part in the endless Cycle.

The Warrior of Light's name is Mid, and he looks just like Cid.
In Dissidia, it is hinted that Cid made the Wo L in his likeness, just as the State made Cosmos in the image of Cid's wife. A recurring name for Cid's sons and grandsons in the series is Mid, making it an apt name for this son of Cid.

The party in FF 1 consisted of the Warrior, White Mage, Thief, and Black Mage.
Based entirely off the fact that the former job class was a major character in Dissidia, while the latter three were unlockable characters in Mario Hoops and Mario Sports Mix.

The Lufenians built multiple WarMECHs to defend the Flying Fortress.
It makes sense: if you could encounter two WarMECHs after taking down one (or running from it), wouldn't that mean there are actually multiple machines of death guarding the Flying Fortress?

Matoya is an incarnation of Ravel Puzzlewell
Both of them are insane, blind, elderly witches who you encounter inside of their hideout, and all of these basic characteristics are found among the other incarnations of Ravel that we see in Planescape: Torment. As stated by Word of God, Ravel's existence spreads through the planes like branches of a tree. Since all universes are intertwined and exist as planes within the Planescape setting, the world of Final Fantasy I would be just as likely as any other plane to host one of her incarnations.