And if so, was there a story that is credited with “inventing” the idea, or did it sort of arise spontaneously? Was much made of the innovation at the time? And what kind of social/literary factors influenced the development?
In modern literature, one form that draws most directly from the fountainhead of mythology, and does it consciously and deliberately, is the heroic romance, which is a form of high fantasy. The world of heroic romance is, as Professor Northrop Frye defines the whole world of literature in The Educated Imagination, “the world of heroes and gods and titans…, a world of powers and passions and moments of ecstasy far greater than anything we meet outside the imagination.”
If anyone can be credited with inventing the heroic romance as we know it today — that is, in the form of a novel using epic, saga, and chanson de geste as some of its raw materials — it must be William Morris, in such books as The Wood Beyond the World and The Water of the Wondrous Isles. Certainly Morris showed the tremendous strength and potential of the heroic romance as an artistic vehicle, which was later to be used by Lord Dunsany, Eric Eddison, James Branch Cabell; by C. S. Lewis and T. H. White. Of course, heroic romance is the basis of the superb achievements of J. R.R. Tolkien.
"High fantasy" as defined by Alexander in his '71 essay defines the fictional secondary world which is discrete from the world we know; this is contrasted with "low fantasy" where the two worlds coexist in some fashion—Narnia can be accessed through a wardrobe, for example, and the Hyborian era of the Conan the Barbarian tales of Robert E. Howard is set in a fantastical prehistoric past of our present era, as outlined in his essay The Hyborian Age (1936).
The first instance of the idea is problematic. The idea of a literary definition of a place which was not the Earth we know in supernatural literature goes back to prehistory; Heaven and Hell in particular have seen their various permutations, sometimes very elaborate ones such as Dante Aligheri's Divine Comedy (1308-1320) and John Milton's Paradise Lost (1667), which were often distinctly their own geographies but tied to this world not geographically, but spiritually. Allegory also had its role, as in John Bunyan's The Pilgrim's Progress (1678), and satire in Samuel Butler's Erewhon (1872). One of the first truly imaginary worlds in the contemporary period was "Allestone," created by the young Thomas Williams Malkin and published in A Father's Memoirs of His Child (1806), although it's difficult to say how much influence this had.
Most of the "high fantasy" concept as we know it today has its immediate roots in the literature of the mid-to-late 19th century, where we begin to see a lot of "low fantasy" involving people entering or accessing imaginary worlds, often through dreams or some form of astral or psychic projection—such as Rudyard Kipling's "The Brushwood Boy" (1895), Lord Dunsany's A Dreamer's Tales (1910), Edgar Rice Burrough's A Princess of Mars (1912), E. R. Eddison's The Worm Ouroboros (1922), etc.—the actual device of displacement is not a new one; Samuel Clemens had used a similar method in A Connecticut Yankee in King Arthur's Court (1889). The device of transition allows the protagonist to act as a surrogate for the audience, to explore these strange worlds with fresh eyes as an explorer would—and not without reason, this was an age of colonialism and exploration, and I touch on this in a discussion of the related Lost World/Lost Race genre.
That being said, you can make a serious argument (as Alexander does) for William Morris' The Wood Beyond the World (1894) as being the first major imaginary world. This is because Morris is specifically not including any transition point; its innovation of just starting us in the imaginary world at the beginning, without any device to "bridge" the reader, is tied in part to its derivation from medieval literature that was heavy with allegory—Morris' unnamed world is a pastiche and a homage to medieval literature, but he doesn't try to set it in an established geographical or historical period or context because he wasn't going for even a veneer of historical accuracy—this is a romanticized medieval setting; u/pipkin42 and I talk a bit about that in Were medieval times always romanticised, or are phrases like "Knight in shining armour" relatively new? Compare that with The Gods of Pegāna (1905) by Lord Dunsany, which builds an entire mythology (but is fairly loose with the geography).
Was much made of the innovation at the time?
It didn't catch on immediately, and I hesitate to claim that Morris was a direct influence on a lot of the immediate examples you get in the early 20th century, since it seems an idea that was developed quasi-independently by at least a couple of different writers. Part of the issue is that Morris gave very limited development to his world over multiple works, in stark contrast to serial fiction like the Oz books of Frank Baum or pulp fiction. Writers like Edgar Rice Burroughs, H. P. Lovecraft, and Robert E. Howard built and developed settings and internal mythologies for their works which made them more attractive to readers, and sometimes these served as "launching points" for high fantasy imaginary worlds. For example, Clark Ashton Smith had previously created "low fantasy" stories which he set in the imaginary worlds/settings of Averoigne, Hyperborea, Poseidonis, Zothique, even Saturn...and then he wrote stories set completely apart from all that, in the alien world of Xiccaraph, "The Maze of Maal Dweb" (1933) and "The Flower-Women" (1935). I won't say this quickly became a standard, but it was established as not-uncommon by at least the first Nehwon stories about Fafhrd and the Gray Mouser appeared in '39.
Most of the time when you think "high fantasy," however, you think of J. R. R. Tolkien and Middle Earth, as detailed in The Hobbit (1937) and The Lord of the Rings (1954). This...really didn't start out as explicitly a high fantasy story, as chronicled by Christopher Tolkien in The History of Middle Earth: Tolkien was trying to create a sort of "mythic past" for England, rudely similar to Robert E. Howard's Hyborian Age, and that basic conception stayed on until fairly late in the writing process—but the final product doesn't create the bridge between this world and Middle Earth (although there are certainly parts of the Silmarillion and the appendices of LOTR where you can see it sort of suggests it). So as published Middle Earth is high fantasy by dint of not mentioning this work—and of course went on to be hugely successful and inspired many subsequent works of fantasy, which followed the same vibe (inspired in part by the concurrent re-popularization of other early fantasy writers through the Ballantine Adult Fantasy series and the Robert E. Howard Sword & Sorcery boom in the late 60s/70s).
Which I hope helps to answer your question.
(A caveat: this all pertains to English literature, I'm not well-versed enough to speak definitively on that in other languages and me secondary sources for those are slim.)