Teach me half the gladness That thy brain must know,
Now that he knows he can't match up to the skylark's pure beauty, the speaker asks the skylark to teach him.
He wants just a piece, just half of the happiness (the "gladness") that he figures the bird must feel.
Such harmonious madness From my lips would flow
Then, he imagines, if he knew the bird's happiness, he would be able to speak a kind of "harmonious madness." This is a key phrase, and it's also an oxymoron. Usually we think of harmony and madness as being a contradiction, but here the speaker buts both together.
The speaker is trying to imagine a kind of song, a kind of poem, that would push beyond the normal human limits, that would allow him to feel and write and sing as purely as a skylark. Again, this idea that the beauty of the bird's song is somehow disorienting and dazing pops up. It's not just harmony that the speaker imagines. It's also madness, too.
The world should listen then, as I am listening now.
This is all he wants: for the world to hear him as clearly as he hears the skylark.
He wants them to absorb his words, to be as entranced with them as he is with the song of the bird.
There's something sweet and sad about this ending. He starts out talking to the bird, asking it questions, and winds up almost jealous of its beauty and its immense power, realizing that he will never know its silent, hidden secrets.