At the end of the Return of the King, why did Aragorn tell the four hobbits
"you bow to no one"?
Does this scene appear in the book?
A similar scene does happen in the books, in the chapter The Field of Cormallen, where after Frodo recovers consciousness following his rescue from Mordor, we have a celebration:
As they came to the opening in the wood, they were surprised to see knights in bright mail and tall guards in silver and black standing there, who greeted them with honour and bowed before them ... And as the Hobbits approached swords were unsheathed, and spears were shaken, and horns and trumpets sang, and men cried with many voices and in many tongues:
'Long live the Halflings! Praise them with great praise!
Cuio i Pheriain anann! Aglar'ni Pheriannath!
Praise them with great praise, Frodo and Samwise!
Daur a Berhael, Conin en Annun! Eglerio!
And so the red blood blushing in their faces and their eyes shining with wonder, Frodo and Sam went forward and saw that amidst the clamorous host were set three high-seats built of green turves ... On the throne sat a mail-clad man, a great sword was laid across his knees, but he wore no helm. As they drew near he rose. And then they knew him, changed as he was, so high and glad of face, kingly, lord of Men, dark-haired with eyes of grey.
And then to Sam's surprise and utter confusion he bowed his knee before them; and taking them by the hand, Frodo upon his right and Sam upon his left, he led them to the throne, and setting them upon it, he turned to the men and captains who stood by and spoke, so that his voice rang over all the host, crying:
'Praise them with great praise!'
It seems evident that Jackson worked elements from this scene into Aragorn's coronation scene for the movies, although the movie scene is considerably less grandiose than that from the book.
As for reasons why, they should be obvious:
In the original book, The Return of the King, the only person who bows during the ceremony is Aragorn, to receive the crown.
Then Frodo came forward and took the crown from Faramir and bore it to Gandalf; and Aragorn knelt, and Gandalf set the White Crown upon his head, and said: 'Now come the days of the King, and may they be blessed while the thrones of the Valar endure!' But when Aragorn arose all that beheld him gazed in silence, for it seemed to them that he was revealed to them now for the first time. Tall as the sea-kings of old, he stood above all that were near; ancient of days he seemed and yet in the flower of manhood; and wisdom sat upon his brow, and strength and healing were in his hands, and a light was about him.
And then Faramir cried: 'Behold the King!' And in that moment all the trumpets were blown, and the King Elessar went forth and came to the barrier, and Húrin of the Keys thrust it back; and amid the music of harp and of viol and of flute and the singing of clear voices the King passed through the flower-laden streets, and came to the Citadel, and entered in; and the banner of the Tree and the Stars was unfurled upon the topmost tower, and the reign of King Elessar began, of which many songs have told.
Oh "it never cease to amaze me,the courage of hobbits".This said in the movie scene of 'The Hobbits'by one of dwarves when he was giving Bilbo the pep talk to enter the dragon's lair.Bilbo,frodo and the other Hobbits had no business being involved in the war of Gandalf and the rest,yet they volunteered and nothing is stronger than the heart of a volunteer.