Help me parse this sentence in The Silmarillion

The word he in your passage is referring to Morgoth himself. Rumour that the Elven princes had designs had come to Morgoth, but he didn’t know what those designs were.

Here’s the longer passage for context:

        They passed through all perils, until they came with the dust of their long and weary road upon them to the drear dale that lay before the Gate of Angband. Black chasms opened beside the road, whence forms as of writhing serpents issued. On either hand the cliffs stood as embattled walls, and upon them sat carrion fowl crying with fell voices. Before them was the impregnable Gate, an arch wide and dark at the foot of the mountain; above it reared a thousand feet of precipice.

        There dismay took them, for at the gate was a guard of whom no tidings had yet gone forth. Rumour of he knew not what designs abroad among the princes of the Elves had come to Morgoth, and ever down the aisles of the forest was heard the baying of Huan, the great hound of war, whom long ago the Valar unleashed. Then Morgoth recalled the doom of Huan, and he chose one from among the whelps of the race of Draugluin; and he fed him with his own hand upon living flesh, and put his power upon him. Swiftly the wolf grew, until he could creep into no den, but lay huge and hungry before the feet of Morgoth. There the fire and anguish of hell entered into him, and he became filled with a devouring spirit, tormented, terrible, and strong. Carcharoth, the Red Maw, he is named in the tales of those days, and Anfauglir, the Jaws of Thirst. And Morgoth set him to lie unsleeping before the doors of Angband, lest Huan come.

Even that is but the prose condensation of the original tale in verse from the Lay of Leithian. It’s a lot clearer in the original. I’ve set in bold the portion of the verse that most directly corresponds to the bold portion you cited from the prose synopsis above:

        Thus Bauglir earned the furrowed scar
that his dark countenance doth mar,
and thus his limping gait he gained;
but afterward profound he reigned
darkling upon his hidden throne;
and thunderous paced his halls of stone,
slow building there his vast design
the world in thralldom to confine.
Wielder of armies, lord of woe,
no rest now gave he slave or foe;
his watch and ward he thrice increased,
his spies were sent from West to East
and tidings brought from all the North,
who fought, who fell; who ventured forth,
who wrought in secret; who had hoard;
if maid were fair or proud were lord;
well nigh all things he knew, all hearts
well nigh enmeshed in evil arts.

        Doriath only, beyond the veil
woven by Melian, no assail
could hurt or enter; only rumour dim
of things there passing came to him.
A rumour loud and tidings clear
of other movements far and near
among his foes, the threat of war
from the seven sons of Fëanor,
from the far Falas, from Fingon still
gathering his armies under hill
and under tree in Hithlum’s shade,
these daily came. He grew afraid
amidst his power once more; renown
of Beren vexed his ears, and down
the aisléd forests there was heard
great Huan baying.

                                Then came word
most passing strange of Lúthien
wild-wandering by wood and glen,
and Thingol’s purpose long he weighed,
and wondered, thinking of that maid
so fair, so frail. A captain dire,
Boldog, he sent with sword and fire
to Doriath’s march; but battle fell
sudden upon him: news to tell
never one returned of Boldog’s host,
and Thingol humbled Morgoth’s boast.
Then his heart with doubt and wrath was burned:
new tidings of dismay he learned,
Sauron was o’erthrown, his strong isle
broken and plundered, how with guile
his foes now guile beset; and spies
he feared, ’till each Orc to his eyes
was half suspect. Still ever down
the aisléd forests came renown
of Huan baying, hound of war
the Valar unleashed in Valinor.

        Once had Morgoth Huan’s fate bethought
long-rumoured, and in dark he wrought.
Fierce hunger-haunted packs he had
that in wolvish form and flesh were clad,
but demon spirits dire did hold;
and ever wild their voices rolled
in cave and mountain where they housed
and endless snarling echoes roused.
From these a whelp he chose and fed
with his own hand on bodies dead,
on fairest flesh of Elves and Men,
’till huge he grew and in his den
no more could creep, but by the chair
of Morgoth’s self would lie and glare,
nor suffer Balrog, Orc, nor beast
to touch him. Many a ghastly feast
he held beneath that awful throne,
rending flesh and gnawing bone.
There deep enchantment on him fell,
the anguish and the power of hell;
more great and terrible he became,
with fire-red eyes and jaws aflame,
with breath like vapours of the grave,
than any beast of wood or cave,
than any beast of earth or hell
that ever in any time befell,
surpassing all his race and kin,
the ghastly tribe of Draugluin.

        Him Carcharoth, the Red Maw, name
the songs of Elves. Not yet he came
disastrous, ravening, from the gates
of Angband. There he sleepless waits;
where those great portals threatening loom
his red eyes smoulder in the gloom,
his teeth are bare, his jaws are wide;
and none may walk, nor creep, nor glide,
nor thrust with power his menace past
to enter Morgoth’s dungeon vast

        Now, lo! before his watchful eyes
a slinking shape he far descries
that crawls into the frowning plain
and halts at gaze, then on again
comes stalking near, a wolvish shape,
haggard, wayworn, with jaws agape;
and o’er it batlike in wide rings
a reeling shadow slowly wings.
Such shapes there oft were seen to roam,
this land their native haunt and home;
and yet his mood with strange unease
is filled, and boding thoughts him seize.

        ‘What grievous terror, what dread guard
hath Morgoth set to wait, and barred
his doors against all entering feet?
Long ways we have come at last to meet
the very maw of death that opes
between us and our quest! Yet hopes
we never had. No turning back!’
Thus Beren speaks, as in his track
he halts and sees with werewolf eyes
afar the horror that there lies.
Then onward desperate he passed,
skirting the black pits yawning vast,
where King Fingolfin ruinous fell
alone before the gates of hell.

The entire Lay is much longer than this tiny excerpt. It sheds much light on things that are sometimes hard to make out in the abridgement that found its way into the published Silmarillion.

It's easy to get tripped up by verbose wording, and this is pretty much the verbosest!

Rumour of he knew not what designs abroad among the princes of the Elves had come to Morgoth

The phrase "he knew not what" is something of an idiom -- which is the only reason why a reader might recognize it and decipher this mess. It means that there was a rumor of something he was not immediately aware of. It was a rumor about "designs abroad among the princes of the Elves", where "designs" would most likely be interpreted as "schemes" or "plots", and "abroad" means that these schemes were spreading between the "princes".

All this had "come to" Morgoth -- reached his ear.