There is no canonical answer to this question
In no published source that touches on the eladrin, and there really aren't that many, is there a canonical answer pre-4E (and 4e ones are just high elves). The celestials have always gotten less attention and detail than the fiends, for obvious reasons. We actually don't have a firm origin, or even suggested origins, for any of them. Notes on the eladrin identify the notion that they might be elven spirits but also reject it as an unfounded rumor - and not in the usual way of winking at the reader and saying "but maybe..."
Here's what we do know
The eladrin originated from a time known as the Age Before AgesFC1. beings of chaos and good who allied with the obyrith known as the Queen of Chaos in the ancient war against the forces of Law. When Miska the Wolf-Spider was banished by the Rod of Law2E:RoSP,3E:FC1 and the forces of Chaos were sundered, the eladrin struck out at their former allies and so devastated the obyriths that they were unable to fight back against a similar betrayal by the tanar'ri.
The eladrin are led by Queen Morwel, who resides in the Court of Stars (both a demiplane attached to Arborea as well as the metonym for the eladrin "leadership" and nobility). She has been the only eladrin leader for as long as anyone is awareBoED, although her consorts have changed over the millennia.
Ruins found in Mithardir (also known as Pelion) as well as encounters with a mysterious and rare race of albinistic "elves" in and around said ruins suggest that the eladrin were either originally very different from what they are now or else were preceded by beings who wielded great and terrible powers, and that whoever erected those now-ruined buildings likely had a hand in the foundation of the cosmos as known2E:DG. Certainly legends speak of a time when Mithardir was forested and a race of titanic beings, perhaps lost deities, ruled the now-dead layer of the planeMotP.
Source List
2nd Edition
Dead Gods (Planescape adventure); Planes of Chaos (Planescape accessory); Planescape Campaign Setting; Planescape Monstrous Compendium II; Rod of Seven Parts (D&D adventure); Warriors of Heaven
3rd Edition
Book of Exalted Deeds; Fiendish Codex I: Hordes of the Abyss; Manual of the Planes; Monster Manual; Planar Handbook
Not an exhaustive list, provided mainly for corroboration of major points or for further reading on topic of interest.
5E Realms lore does not specifically address this. 4E Realms lore, on the other hand, does. Their exact origin is subject to some debate, even among the elves.
Collectively, the elves are known as the Tel’Quessir6 ("The People"), a title that encompasses Eladrin, Wood Elves, High Elves, Sea Elves, and so on. It is 'widely accepted' that the Tel’Quessir are native to the Feywild1. It is believed that they first immigrated to Abeir-Toril over a hundred millenia ago, while the exact date is uncertain it is believed to have been prior to −24,000 DR, with some manuscripts indicating it was as early as −30,000 DR, the same time in which dragons were setting up their empires (Note: Source on this is from AD&D)4. The Wild Elves were the first to arrive.1
The exact origin of the Elvish race is subject to some debate. Some ancient manuscripts suggest that they were the result of a battle between Gruumsh and Correlon that occurred in the Feywild. During said battle, Correlon was injured and his shed blood became the Eladrin.2. Other types of elf, not liking the superiority this implies in that the Eladrin are the first and 'purest' of elves, disagree with this theory3. Another theory postulated is that since the Feywild tends to be a 'reflection' of the Prime Material Plane, and thus produces creatures that are echoes of creatures from the Prime, it is also possible that the Elves are one such echo... though what, exactly, they would be an echo of is uncertain1.
Perhaps also worthy of mentioning is the current in-lore explanation for how the Feywild (also known as Faerie) is treated according to current lore. The Feywild was created as it exists today, a 'lighter' echo of the Prime Material Plane. It was made by Primordials in the far distant past, before recorded history begins.5
For a while, the Feywild existed alongside the Prime and interaction between the two was common. But, at some point in ancient history, the two drifted apart and travel between them became increasingly difficult2. This is the explanation for why, in prior editions of D&D, the Feywild didn't "exist" in the Forgotten Realms. It was out there, you just couldn't get to it.
The Spellplague (the event that 'kicks off' 4E in the Realms) changed that, reshuffling the cosmology to drag the Feywild back into close proximity, and placing it in the accessible location it resides in today5.
Checking through published 5E Realms material, it does not contradict any of this... the 5E Realms cosmology is essentially the same as the 4E Realms cosmology, and the lore established within 4E is considered to hold true in 5E, except where directly overridden. And even then, the changes are usually justified in-lore (such as the array of gods and goddesses getting rejiggered due to Ao meddling with them).
The primary lore sources we have in 5E Realms are the Sword Coast Adventurer's Guide, the DMG, and published adventures. None of these go into detail on the origins of the Elves, or even on the specific nature of the Feywild... thus we may assume the lore of older sources hold true.
Sources
1 Brian R. James and Ed Greenwood (September, 2007). The Grand History of the Realms. (Wizards of the Coast)
2Bruce R. Cordell, Ed Greenwood, Chris Sims (August 2008). Forgotten Realms Campaign Guide. (Wizards of the Coast)
3James Wyatt (December 2007). Dragon 361: A Fractured Family. Wizards of the Coast.
4Steven E. Schend and Kevin Melka (1998). Cormanthyr: Empire of the Elves. (TSR, Inc) NOTE: AD&D source
5Dungeon Master's Guide 4th edition. (Wizards of the Coast)
6Kim Mohan ed. (2015). Sword Coast Adventurer's Guide. (Wizards of the Coast)
Best Answer
There are several fundamental planes: the Astral Sea, the Elemental Chaos, the Mortal World, and the World's two echoes: the Feywild (from which the Eladrin came), and the Shadowfell. And then there's Sigil, the city of doors, a plane which doesn't quite fit in anywhere and may not rightly exist inside this particular cosmology at all.
There's also the Far Realm, which exists outside all of that.
Wizards provided a marvellous map of the planes in this excerpt on the planes. It contains a lot more detail of what's in the planes, but you can glaze over that for now and focus in the big bits:
A full understanding of the planes is inseperable from how they came to be. Bear in mind, all history of the default Points of Light setting is (mostly) from the perspective of living mortals, so all the details are fuzzy and there's a couple of versions of everything. That's deliberate and works to your benefit as DM: you have plenty of room to make stuff up and come up with alternatives. The important parts of the history are clear, though, assuming they're not lies.
A Brief History of Life, the Universe and Everything
Citations in this section are from the Manual of the Planes (abbreviated to MOP), the Dungeon Master's Guide (DMG), and The Plane Below (TPB), the sourcebook for detail on the Elemental Chaos. (For further reading there's also The Plane Above for the Astral Sea, and specific sourcebooks like The Shadowfell and Underdark, but I'm not quoting those here.)
The Astral Sea and Elemental Chaos
Once upon a time, there was only the Elemental Chaos and the Astral Sea. Nobody's sure how these came about, because this happened before all mortal or immortal life.[DMG 160] In the former arose the Primordials, the embodiments of Chaos.[DMG 160, MOP 70] In the latter arose the Astral Sea's version of the Primordials: the Gods you probably know well, and the embodiments of Order.[DMG 160]
Birth of the Mortal World and its Echoes
Eventually, the Primordials wanted to create something altogether new. They took some of the infinite stuff of the Elemental Chaos and crafted the elements into a great, spherical world - the Mortal World in its earliest form was born, then still just as hostile and chaotic as the Elemental Chaos itself.[MOP 70] The Primordials further stripped away the brightest parts of the world, and the darkest shadows of the world, and cast them aside. They would eventually recoalesce: the brightest parts gathered together and became the Feywild, and the darkest parts became the Shadowfell.[DMG 161]
Later, the Gods would take notice of this experiment. They visited the Mortal World and stabilised it, separating it into seas and land masses, and sprinkling it with life, making it what we know now.[MOP 70] The Primordials eventually discovered this, found their work of art desecrated with Astral life and stability, and wished to return it to the chaotic thing it was before.[MOP 70] Thus the Dawn War was fought between the Gods and Primordials over the fate of the World.[MOP 70] The Gods won, and the Primordials who survived the conflict were imprisoned or scattered far throughout the Elemental Chaos.[TPB 65]
The Far Realm
The three planes, however infinitely large they are, are only a small bubble in a greater and unknown cosmology. Beyond all we know lies the Far Realms, the origin of many aberrant creatures who are strange and alien and generally seem to want us dead or just make us end up that way anyway.[MOP 30] Please do think of Lovecraft, and think of it as only the least of what the Far Realm may have to offer.
If there's anything that actually knows how the planes came to be, it's probably from the Far Realms, but it's unlikely you'd be able to communicate very effectively with whatever ancient thing witnessed the birth of the cosmology itself.
Sigil
This demiplane serves a role as hosting many secret gateways between the various locations in the 4e cosmology. Elsewhere in a comment on a now-deleted answer, Wesley Obenshain summarised this plane very well (and with more knowledge than I have about it):
More info can be found in the Manual of the Planes on page 25.
How are they connected?
Sigil, the City of Doors, is connected to assorted doors everywhere.
The Mortal World, Shadowfell and Feywild are intimately connected. Wander through the brightest, wildest places in the Mortal World and you may find yourself stumbling into the Feywild,[MOP 34] and likewise for the deepest or darkest places.[MOP 50] Connections come and go, and people do get lost there, just as things there may get lost here.
The Mortal World is not as well connected to the Elemental Chaos or Astral Sea, though. One can stumble across connections to the Elemental Chaos in deeply elemental places still,[MOP 64] but connections to the Astral Sea are rare.[MOP 86]
Of course, creatures who know the proper rituals, possess the proper powers, or who know where the breaches between the planes can be found, can easily traverse the realm. There are several of these methods in the Manual of the Planes.
Fan theories
There's a few, but here's two. The nice thing about how D&D 4e's lore is handled is that fan theories can pretty easily be adopted as truth in your campaign's lore.
The Shadowfell and Feywild: our past and future
The Feywild and Shadowfell may represent the past and future of the Mortal World: the former is the World an eternity ago when it was young and teeming with life, while the latter may be the World's distant future, when it is ancient, barren and dying.
The planes are an impurity in the cosmos
It's possible that rather than aberrant creatures from the Far Realm being the intruders, we are. Our small bubble of creation may actually be an impurity and infection in the greater cosmos, and the aberrant creatures are the white blood cells working to cleanse it.
(Both of these I picked up from another of our members, BESW.)