Aragorn fell to his knees with that fearful and saddened look on his face because he knew that he couldn't save Minas Tirith without the aid of the Men of the Mountains; Sauron's victory was assured if Aragorn couldn't get the Army of the Dead to fulfil their oath.
Peter Jackson et al changed the geography of Middle-earth here to suit the film. As described, Aragorn and his companions exit the Haunted Mountain and appear to see Pelargir, the ancient seat of the Sea Kings upon the River Anduin. In the books, they come out some 300 miles to the north-west, in the Morthond Vale, near to the Stone of Erech (see map below, due south of Edoras at top centre).
It was at the Stone that (according to the books) Aragorn gathered the Dead Men (whom Legolas saw following behind them), calling upon them to fulfil their oath:
The hour is come at last. Now I go to Pelargir upon Anduin, and ye
shall come after me. And when all this land is clean of the servants
of Sauron, I will hold the oath fulfilled, and ye shall have peace and
depart for ever. For I am Elessar, Isildur’s heir of Gondor.
From The Return of the King (The Passing of the Grey Company)
The Grey Company (Aragorn's kinsmen, fellow Dunedain of the North) and the Sons of Elrond were also with them as they passed through the mountains, and they still had a very long way to ride through Southern Gondor before Aragorn let loose the Shadow Host to defeat the Corsair fleet ("Come! By the Black Stone I call you!"), then man the ships and come to the aid of Minas Tirith.
It is also interesting that Aragorn released the Men of the Mountains once they had liberated Pelargir, before setting sail for the White City.
Hear now the words of the Heir of Isildur! Your oath is fulfilled. Go
back and trouble not the valleys ever again!
From The Return of the King (The Last Debate)
In the books, the ships were manned by men of Gondor from the southern regions and freed captives of the Corsairs, and not the ghostly deserters.