Matthew Arnold himself describes the scene at Dover, and you can find that description in the very first stanza of the poem itself. It seems that Arnold is probably looking out his window, at night, viewing the moonlight that is falling on the beach near his dwelling. The moon is out, the air smells sweet, and the ocean waves are lapping at the stones on the beach.
As for the poetic meaning behind this scene, he uses poetic techniques to more descriptively convey his message, and ties the scene to the underlying current of melancholic emotion that he is feeling as he ponders the scene. He uses imagery, or the 5 senses, to relay the scene in a descriptive manner; he describes the "grating roar" of the waves on the pebbles, the sound of the waves as a "cadence", the ocean and cliffs as a "gleam" and "glimmering", the land as "moon-blanched." All of these paint a picture that helps the reader to feel they are right there with him. He uses personification (giving inanimate objects human-like traits); the cliffs "stand", the moon "lies", the sea "meets", and the waves "roar". All of this makes the ocean, moon and scene as a whole seem like a living entity, making it more real and imaginable.
Another aspect of the poetic meaning of the scene is how Arnold tinges his description of the scene with his own personal viewpoint, his own melancholy and depressed mood at the time. Here is a beautiful beach scene in front of him, but he sees it only as a token of sadness. He feels that the ocean brings "the eternal note of sadness" in, instead of a calming and soothing note. He feels the scene rings perfectly of the "turbid ebb and flow of human misery." Taking a scene, and infusing it with the emotion and meaning of the moment is a poetic technique that makes an ordinary scene more symbolic and profound. Arnold, disillusioned about his world because of the war and suffering he has seen in it, takes the scene before him and ties it all back to his hopelessness. A poet is able to take ordinary images and experiences and give them a symbolic undertone that adds more depth to them, and that is what Arnold has done here.
I hope that those thoughts help a bit; good luck!