The poet cannot see the beach but can only hear the sound made by the waves breaking on the strand and rattling all the pebbles as they race over them and then rattle them again as they retreat. It is not a nice, sandy beach but a barren strand covered with rocks and pebbles. The sound reminds the poet of something he read in Sophocles, the ancient Greek tragedian who is best known for his play Oedipus Rex. According to Arnold:
Sophocles long ago
Heard it on the Agaean, and it brought
Into his mind the turbid ebb and flow
Of human misery;
With regard to the question of how Arnold compares the modern world with a shingled beach, the sound reminded Sophocles of one fact about humanity and reminds Arnold of a related fact.
Find also in the sound a thought,
Hearing it by this distant northern sea.
What the ghostly sound of the pebbles brings to Arnold's mind, along witn the remembrance of Sophocles, is the thought that the Sea of Faith is retreating with
Its melancholy, long, withdrawing roar,
Retreating, to the breath
Of the night-wind, down the vast edges drear
And naked shingles of the world.
By the Sea of Faith he means the established Christian religious faith which had provided spiritual solace for the people of the Western world for well over a thousand years. That faith was being undermiined by scientific discoveries and scientific theories. The established churches depended on the Bible, and the teachings of the Bible were being discredited by scientific enlightenment. The account of the origin of the universe and of mankind given in Genesis was regarded by many thinking people, including Arnold himself, as nothing but mythology; and this was bringing the entire Bible and the entire edifice of Christianity into question.
Arnold seems to be viewing the barren strand and the cold, dark English Channel as the vast, empty world without religious faith--a materialistic world which
Hath really neither joy, nor love, nor light,
Nor certitude, nor peace, nor help for pain
That was the modern godless existential world as he saw it in the middle of the nineteenth century. That is the view reflected in so much modernist and post-modernist literature ever since. The twentieth-century wouid be a century of unprecedented chaos, not unrelated to the retreat of the "Sea of Faith."