The purpose of "To His Coy Mistress" is an appeal to seize the day. Specifically, the speaker tries to convince his beloved to make love to him while they still can.
The speaker begins by noting the lady's coy personality. He insists that they simply don't have time to engage in games of pursuit. In an ideal world, he comments, they could do things her way. There would be time for them to sit and talk, and he could spend a hundred years simply admiring her eyes.
Yet this isn't an ideal world, and the speaker asserts that time is not on their side. She could die at any time, at which point she would be in the grave, past all of life's pleasures, and in the loathsome company of worms. This is a morbid image, likely intended to compel the young lady to give in to his desires.
After presenting a series of death-related images, the speaker then contrastingly offers images of the lady's youth and beauty. He reminds her that she still holds the "dew" of youth and is vibrantly alive in this moment. Therefore, she should seize this opportunity to "sport" with the speaker.
Overcome with desire for this coy lady, the speaker's purpose is to convince her to overcome her hesitation and to make love to him. He claims that by doing so they will be living full and pleasurable lives, despite the knowledge that they must eventually face mortality.