My heart aches, and a drowsy numbness pains My sense, as though of hemlock I had drunk,
The speaker says that his heart hurts as if he has just drunken poison.
"Hemlock" is the poison that the Greek philosopher Socrates took when he was put to death for corrupting the youth.
The speaker feels woozy and numb, like when the dentist puts you on Novocain. Imagine him swaying back and forth, kind of drunk and out of it.
The "ache" in his heart almost sounds pleasurable, the way he describes it. Like when you hear a sad song you really love that just pierces your heart, and you're like, "This makes me so sad!" but if anyone tried to turn it off you'd throttle them. Like that.
Or emptied some dull opiate to the drains One minute past, and Lethe-wards had sunk:
OK, so maybe "poison" is a bit exaggerated. He's not dying, after all. He tries another approach to explain how he feels.
He feels as though he has drank some powerful drug or painkiller ("opiate") that causes him to "sink" into a kind of oblivion.
In Greek mythology, "Lethe" was a river in Hades (the Underworld) that made people forget all their memories if they drank from it.
There's really no way to dance around it: the speaker is comparing his feeling to being totally strung out on drugs.
"Opium" is a powerful drug made from the poppy flower, and it was all the rage among certain adventurous types in the 19th century. The poet Samuel Taylor Coleridge, for example, was an opium addict, as was the writer Thomas de Quincy, who wrote an essay titled, "Confessions of an Opium Eater." This was before people discovered just how toxic opium is for the body.
'Tis not through envy of thy happy lot, But being too happy in thine happiness,—
Now we know that the speaker must be addressing the nightingale of the title.
He wants to clarify that the pain he feels is not because he is jealous of the bird's happiness. Instead, he is excessively happy for the bird's happiness. He's like that friend who bursts into tears when you share really good news and cries, "I'm just . . . so . . . happy . . . for you!" — but you're not sure if they are really happy for you or just sad for themselves.
That thou, light-winged Dryad of the trees, In some melodious plot Of beechen green, and shadows numberless, Singest of summer in full-throated ease.
And why is the nightingale so happy? Because it gets to sit in the trees all day and sing about summertime. It's to the trees what Jimmy Buffet is to the beach (hey-ya!).
The nightingale is not a large bird, and it can fly, which seems like enough grounds to call it "light-winged" (which is pronounced with three syllables, by the way).
And in Greek mythology, a "dryad" is a nymph (female spirit) that lives in the trees.
The bird makes whatever space or "plot" it inhabits "melodious," and this particular plot seems to have beech trees, giving it a "beechen green" color.
The nightingale doesn't hold back. It sings with a "full throat," like an opera singer in a solo. We imagine that this poem takes place in the peak of summer.