Nature’s first green is gold,
Her hardest hue to hold.
The first line equates the color green to gold, signifying its value. The second line acknowledges that the green hue is the hardest to keep due to the transience of youth and beauty. In essence, anything beautiful is short-lived.
So Eden sank to grief,
So dawn goes down to day.
These lines utilize a religious allegory to compare the eventual and inevitable loss of beauty and youth to the fall of man in the Garden of Eden. The speaker does not offer an opinion on the cause of this loss, instead simply noting it.
The line "So dawn goes down to day" is particularly interesting because dawn is usually considered a precursor to day, not something that gets subsumed by it. However, here, dawn turning to day is a loss. Sunset would be perhaps the more natural choice to signify loss, but Frost subverts that expectation.
Nothing gold can stay.
This last line, a standalone sentence and the poem's title, contains a syllable fewer than each preceding line, and its abruptness and brevity emphasize the speaker's grief regarding the idea that the line expresses: nothing good lasts forever.