This is as high-flown a Latinate word as the clouds it figuratively evokes (it comes from nubes, a cloud). It means to darken, dim, cloud over, or obscure. It’s not a word to be wasted on everyday conversation, but on its rare outings seems to be the special province of the more ponderous political speeches and newspaper editorials.
Nineteenth-century reviewers used it to suggest that a writer had been less than transparently clear in his exposition, as here in a squib in The Princeton review in 1832 about a book by Samuel Taylor Coleridge: “There is here fine criticism, classic wit, poetic dreaming, and some grains of sound doctrine, but so obnubilated with the fumes of German metaphysics, that we become giddy, and lose all power of comprehension”. And here it appears as an adjective in a put-down of Walt Whitman in the Southern Literary Messenger in 1860: “Here is the sample of his obnubilate, incoherent, convulsive, flub-drub”. They don’t write criticism like that any more (and they don’t spell that last word that way, either, but as flubdub).
Some modern writers have found it useful to suggest the ponderous wordsmithery of previous generations. Neal Stephenson, for example, in his novel Quicksilver set in the seventeenth century:
This was a wonder all by itself, with its ropewalks — skinny buildings a third of a mile long — windmills grinding lead and boring gun-barrels, a steam-house, perpetually obnubilated, for bending wood, dozens of smoking and clanging smithys including two mighty ones where anchors were made ...
And Patrick O’Brian had Dr Maturin say it in The Mauritius Command: “It is the pity of the world, Dr McAdam, to see a man of your parts obnubilate his mind with the juice of the grape.”