If time lasted forever, the speaker and his mistress could court and play games forever. Coyness is equated with flirtation. From lines 1-21, the speaker discusses how he and his mistress would spend such a vast amount of time: from before Noah’s flood until the end of time.
At line 21, he is reminded of their mortality. When she is in the grave, his praises will no longer be heard (lines 26-27), and she eventually will become food for worms and then dust. There is a sexual connotation here with the worms “trying her virginity” as they take her body, which she will not be able to defend. This disturbing image is meant to contrast death with the appreciation of life. He concludes by saying the grave is a private place but none “do there embrace.” We had better embrace now, because we can’t embrace when we’re dead.
Starting with line 33, he begins his direct plea to seize the day. “Now” is repeated to increase the urgency. He asks her to “tear our pleasures with rough strife” “like amorous birds.” In other words, he asks her to love (and make love) with an urgency that suggests they have little time left. He prefers to “devour” time rather than let time slowly devour them.