How we cite our quotes: The poem is 159 lines long, so we just cite by line number.
Quote #4
For I have learned
To look on nature, not as in the hour
Of thoughtless youth, but hearing oftentimes
The still, sad music of humanity,
Nor harsh nor grating, though of ample power
To chasten and subdue. (88-93)
Now the speaker has matured enough from his "thoughtless youth" to see beyond the immediate, physical pleasure in a beautiful scene. He's able to see something wider and more universal ("the still, sad music of humanity"), which helps to "chasten and subdue," or calm him down.
Quote #5
A motion and a spirit, that impels
All thinking things, all objects of all thought,
And rolls through all things. (100-102)
The speaker describes the "presence" that he has learned to detect in nature. It "impels," or animates "all thinking things," and connects everything.
Quote #6
Knowing that Nature never did betray
The heart that loved her (122-123)
The speaker has learned to rely on Nature (now with a capital "N"!) with almost religious devotion. His enjoyment of nature has turned into a kind of pantheism, or natural religion: he's learned to see a divine spirit in everything in nature, and he prays to it.