I'm not entirely certain this means "born on the moon". Most texts that I can find refer to Elfland as being subterranean:
The fairyland of the ballad tradition similarly existed in a kind of wilderness, albiet one suffused in preternatural light. As already indicated, Thomas Rhymer's Elfland was situated in some sort of subterranean locale wherein 'he saw neither sun nor moon / But heard the roaring of the see'. Likewise, the enchanted wood in 'Tam Lin' lacked solar and lunar illumination: 'Seven days she tarried there/Saw neither sun nor meen'.
Frequent mention is made of the location of the home of the elves in the witch trials, virtually every example placing the fairies beside or inside hills. The trial of Lady Fowlis or Katherine Ross in 1590 reported that she 'wald gang in Hillis to speik the elf folk'. In 1615, Jonet Drever was convicted for the 'fostering of ane bairne in the hill of Westray to the fary folk, callit of hir guid nichbouris'. In Shetland, Katherine Jonesdochter, tride in 1616, saw trows on a hill called Greinfaill.
Source: Scottish Fairy Belief: A History; Henderson & Cowan, 40-41
This begs the question, are we even sure what is meant by moon-born? For that, we go to the OED (which actually mentions King Arthur):
moon-born adj. chiefly poet.
(a) born under the moon's influence; (b) originating from the moon.
Source: Oxford English Dictionary: moon-born
Perhaps what is meant here is not "originating from the moon", but born under the moon's influence. This is back up by the else tended to operate at night:
The face of the country, too, might have some effect; as we should naturally attribute a less malicious disposition, and a less frightful appearance, to the fays who glide by moon-light through the oaks of Windsor, then those who haunt the solitary heaths and lofty mountains of the North. The fact at least is certain; and it has not escaped a late ingenious traveller, that the character of the Scottish Fairy is more harsh and terrific than that wish is ascribed to the elves of our sister kingdom.
A beautiful reason is assigned by Fletcher for the fays frequenting streams and fountains: He tells us of
A virtuous well, about whose flowery banks
The nimble-footed Fairies dance their rounds,
By the pale moon-shine, dipping oftentimes
Their stolen children, so to make them free
From dying flesh and dull mortality.
It is sometimes accounted unlucky to pass such places, without performing some ceremony to avert the displeasure of the elves.
Source: The Book of Scottish Ballads, Etc.; Alexander Whitelaw; 438-439