Fade far away, dissolve, and quite forget What thou among the leaves hast never known,
If this were a movie, now would be the part when the screen gets all blurry, a harp starts playing, and the dream sequence begins.
The speaker dreams of "fading" out of the world, of just disappearing in a very quiet way.
He wants to forget about those things that the nightingale has never had to worry about. Again, we don't know much about which things he means specifically, but we assume they must have to do with the stresses and cares of living in human society.
The bird is free of such cares.
The weariness, the fever, and the fret Here, where men sit and hear each other groan;
Wait, this is supposed to be a dream sequence: why is he talking about these depressing things? It seems just he just can't leave the world behind.
The world is full of tired and "weary" people, sickness ("fever"), and massive stress ("fret"). He reduces all of society down to one depressingly exaggerated image: people sitting around and listen to each other "groan" and complain.
That's a pretty bleak view of the world, but it just goes to show how much of an effect the nightingale has had on him. Compared to the nightingale's carefree song, our voices sound like groans.
Where palsy shakes a few, sad, last gray hairs, Where youth grows pale, and spectre-thin, and dies;
He decides to take the whole depressing images thing to a new level, describing the world as a place where the uncontrollable movements of illness shake the "last gray hairs" on a dying man's head. Palsy is a disease the causes sudden involuntary movements, and so this gray-hair person is no long capable of controlling his own body.
He's also almost bald.
In this section, Keats confronts one of his favorite enemies: time. After you read this poem, check out the "Ode on a Grecian Urn" in which he tries magically to stop time.
Time is the speaker's enemy because it causes young and beautiful people to turn old, "pale," thin as a ghost, and, eventually, dead as a doornail.
Put simply, time = death, death = bad, so time = bad.
Where but to think is to be full of sorrow And leaden-eyed despairs,
The world is a place where any kind of thinking leads to depressing thoughts and worries. There are no thoughts that can ultimately bring joy or peace: thinking itself is the problem.
These sad and "despairing" thoughts make your eyelid like lead weights. You have trouble just staying awake and conscious during the day. The world totally wears people down and tires them out.
Where Beauty cannot keep her lustrous eyes, Or new Love pine at them beyond to-morrow.
The speaker continues to explain why the world of human time is such a bad place. Neither Beauty nor Love can survive there for long.
Beauty loses her glowing ("lustrous") eyes, probably when they become "leaden" from depressed thoughts. Or maybe just from old age.
And new Love cannot fawn ("pine") over Beauty's eyes once they have lost their luster. Love is fickle like that, and, as anyone who has ever been through junior high school knows, it often doesn't last "beyond to-morrow."