Como o exército de Sauron sabia que o Anel foi destruído?

Era o poder de Sauron, intimamente ligado ao do Anel, que os alimentava.

Aqui está o texto real do livro descrevendo o que acontece com o anfitrião no Portão Negro durante os eventos em Orodruin:

But the Nazgûl turned and fled, and vanished into Mordor’s shadows, hearing a sudden terrible call out of the Dark Tower; and even at that moment all the hosts of Mordor trembled, doubt clutched their hearts, their laughter failed, their hands shook and their limbs were loosed. The Power that drove them on and filled them with hate and fury was wavering, its will was removed from them; and now looking in the eyes of their enemies they saw a deadly light and were afraid.

_{-- The Return of the King, Part 2, Chapter 4: The Fields of Cormallen (emphasis mine)}

Além disso, o evento foi tão cataclísmico que seus efeitos foram visíveis a uma grande distância.

Then rising swiftly up, far above the Towers of the Black Gate, high above the mountains, a vast soaring darkness sprang into the sky, flickering with fire. The earth groaned and quaked. The Towers of the Teeth swayed, tottered, and fell down; the mighty rampart crumbled; the Black Gate was hurled in ruin; and from far away, now dim, now growing, now mounting to the clouds, there came a drumming rumble, a roar, a long echoing roll of ruinous noise.

‘The realm of Sauron is ended!’ said Gandalf. ‘The Ring-bearer has fulfilled his Quest.’ And as the Captains gazed south to the Land of Mordor, it seemed to them that, black against the pall of cloud, there rose a huge shape of shadow, impenetrable, lightning-crowned, filling all the sky. Enormous it reared above the world, and stretched out towards them a vast threatening hand, terrible but impotent: for even as it leaned over them, a great wind took it, and it was all blown away, and passed; and then a hush fell.

The Captains bowed their heads; and when they looked up again, behold! their enemies were flying and the power of Mordor was scattering like dust in the wind. As when death smites the swollen brooding thing that inhabits their crawling hill and holds them all in sway, ants will wander witless and purposeless and then feebly die, so the creatures of Sauron, orc or troll or beast spell-enslaved, ran hither and thither mindless; and some slew themselves, or cast themselves in pits, or fled wailing back to hide in holes and dark lightless places far from hope. But the Men of Rhûn and of Harad, Easterling and Southron, saw the ruin of their war and the great majesty and glory of the Captains of the West. And those that were deepest and longest in evil servitude, hating the West, and yet were men proud and bold, in their turn now gathered themselves for a last stand of desperate battle. But the most part fled eastward as they could; and some cast their weapons down and sued for mercy.

_{-- The Return of the King, Part 2, Chapter 4: The Fields of Cormallen (emphasis mine)}

Como sua pergunta tem um clipe de filme, vale a pena examinar o que está acontecendo na versão do filme. O troll e os orcs são distraídos pelos sons horríveis que emanam de Mordor, depois fogem quando eles vêem a Torre de Sauron implodir e as "fundações de Mordor" começam a entrar em colapso.

CLOSE ON: The RING sits on the river of LAVA for a brief moment, then SINKS away . . .

ANGLE ON: the EYE OF SAURON . . .

WIDE ON: Across the GREAT PLAINS - MOUNT DOOM is clearly IMPLODING ...

INTERCUT WITH:

EXT. BLACK GATES OF MORDOR - DAY

On the BATTLEFIELD, all eyes turn to MOUNT DOOM ...

ANGLE ON: The CAVE TROLL, with ARAGORN at his mercy, is distracted by the SOUNDS emanating from MORDOR . . . he turns in TERROR and RUNS OFF . . .

EXT. BARAD-DUR - DAY

ANGLE ON: The FELLOWSHIP turn to watch as the MASSIVE DARK TOWER of SAURON shakes itself to pieces!

TOWERS FALL and WALLS CRUMBLE; vast spires of SMOKE and spouting STEAM billow up!

The FOUNDATIONS explode apart ...

EXT. BLACK GATES OF MORDOR - DAY

The BLACK GATES collapse in a huge cloud of ASH!

THE ORCS are FLEEING in all DIRECTIONS . . . As the very foundations of MORDOR collapse!

_{^{Movie Script - Return of the King - Jackson/Boyens}}

Como outros afirmaram, o Olho soube instantaneamente porque estava literalmente extinto e a torre em que estava estava desmoronada. Isso porque o Olho era a última vida que ele tinha, seu corpo físico havia morrido há muito tempo e sua existência estava ligada ao Anel. Quando foi destruído, Sauron também estava. Além disso, os exércitos sabiam não apenas porque podiam ver e ouvir o que estava acontecendo, mas podiam sentir a presença do Anel e, quando foi destruído, sabiam que não tinham mais esperança porque não havia nada para eles lutar por.

O exército não sabia necessariamente, mas o exército era impulsionado pela vontade de Sauron. Quando ele focou seus pensamentos em outro lugar, no próprio anel momentos antes de ser destruído, seu exército já parou.

And far away, as Frodo put on the Ring and claimed it for his own, even in Sammath Naur the very heart of his realm, the Power in Barad-dûr was shaken, and the Tower trembled from its foundations to its proud and bitter crown. The Dark Lord was suddenly aware of him, and his Eye piercing all shadows looked across the plain to the door that he had made; and the magnitude of his own folly was revealed to him in a blinding flash, and all the devices of his enemies were at last laid bare. Then his wrath blazed in consuming flame, but his fear rose like a vast black smoke to choke him. For he knew his deadly peril and the thread upon which his doom now hung.
From all his policies and webs of fear and treachery, from all his stratagems and wars his mind shook free; and throughout his realm a tremor ran, his slaves quailed, and his armies halted, and his captains suddenly steerless, bereft of will, wavered and despaired. For they were forgotten. The whole mind and purpose of the Power that wielded them was now bent with overwhelming force upon the Mountain. At his summons, wheeling with a rending cry, in a last desperate race there flew, faster than the winds, the Nazgûl, the Ringwraiths, and with a storm of wings they hurtled southwards to Mount Doom.

_{— The Return of the King, Part 2, Chapter 2: "Mount Doom" (emphases mine)}

Além das citações em outras respostas, isso mostra claramente que seus exércitos foram impulsionados pela mente de Sauron, a vontade de Sauron. Eles não sentiram a destruição do Anel em si, eles sentiram a mente de Suaron se retirar como resultado.