Sure, spring's all about the newborn lambs and flowering trees and hatching eggs, but where there's a rebirth, death and decay can't be too far behind. Under those jaunty yellow daffodils are the dead bodies of older daffodils. As hard as this poem celebrates May, it's also about the circle of life.
In fact, that's the speaker's main point in the final stanza: because we're getting older every minute, we might as well have fun while we're still young and beautiful. It's all about seizing the day before you're too weak and ugly.
Line 57: He and Corinna are young and in love right now. There's no point in delaying their fun because they'll never be able to appreciate this much again.
Lines 61-64: The speaker emphasizes how short time is by comparing it to drop of rain. Okay, so you know from 5th-grade discussions of the water cycle that evaporated water isn't actually lost, but chances are, once a drop of water has evaporated, you'll never see that particular bit of H2O again. Similarly, once you die, you never get another life.
Lines 65-66: Notice how Herrick keeps most of his imagery on the lighter side. Instead of dropping in some graves and corpses, he keeps it artistic: one day they'll only exist as stories or poems, not as humans.
Lines 67-68: As Marvell observed in "To His Coy Mistress," "the grave's a fine and private place but none, I think, do there embrace." Once you're dead, all chances for fun, sex, and maypole-dancing are dead too.
Line 69: In this line Herrick describes a contemporary philosophy about life and death: from the moment you're born, you start dying. Pretty depressing, right? It's a death-centric view that makes it all the more urgent to make use of the time you have.