It seems easy enough to explain how most of this applies to
any water-fowl: its fine plumage, unlike some human clothing, doesn't make rustling noises or the like as the swan moves, which it may do on land or on water. Or it may fly high in the air.
in particular? (I've looked in a few places and everyone agrees that this specific thing is the intended answer.) I think there are two clues. The first is in this bit:
My frettings loudly rush and ring
Above the people and most clearly sing
When I forth-fare on air
I'll quote a couple of other translations/paraphrases. First, from here:
wide over the people. My bangles then
loudly jangle, jingling—brightly singing,
when I am not resting upon
the flood or the fold—a faring stranger.
And from here:
My white pinions
resound very loudly, ring with a melody,
sing out clearly, when I sleep not on
the soil or settle on grey waters — a travelling spirit.
And from here which I found the most informative and is the source for much of what I say below:
My adornments then resound loudly, and ring, and sing brightly, when I am not hanging on sea or land, a traveling spirit.
The key point is that the riddle is making a deliberate contrast: the "clothing" of the riddle's subject is quiet on land and on water but resounds, sings, rings, jangles, etc., when it is flying in the air. And
it turns out that in mediaeval times it was thought, or at least said, that the swan's feathers produce a beautiful musical song while it is in flight. (A flying swan's wings absolutely do make a sound, but I don't think it's particularly beautiful or musical.) The last of the pages I linked to above cites an Old English poem called Phoenix, which compares the phoenix's beautiful song favourably with, among other things, "the feathers of the swan", and a letter from Gregory of Nazianzus that I wasn't been able to find but Quuxplusone kindly located in a book (see comments) that has a swan speaking of "when we allow our wings to blow something sweet and harmonious to the western wind".
The second thing that might be relevant is in the second line of the section I quoted above. Here it is in the original Old English:
ofer folc byreð. | Frætwe mine
swogað hlude | ond swinsiað,
torhte singað, | þonne ic getenge ne beom
Those words "swogað" and "swinsiað"
are etymologically related to swan! The swan appears to get its name for its reputation for beautiful singing, and the OED entry for "swan" makes specific reference to that verb "swinsian", meaning to make melody.
I am no Anglo-Saxon scholar, and maybe more is known than this, but I think the above is sufficient to explain why the specific answer everyone gives is surely the intended one.