Mark but this flea, and mark in this, How little that which thou deniest me is; (lines 1-2)
For an unmarried woman in Donne's time, reputation was almost everything. If you lost your reputation, you lost all chance of forming a good marriage and might as well enter a nunnery. So in the first lines, the speaker tries to take this enormous source of anxiety and compact it down to the size of a flea.
Thou know'st that this cannot be said A sin, nor shame, nor loss of maidenhead; (lines 5-6)
The speaker argues that when the same flea bites two people, it is no different from them having sex. If you can't lose your virginity ("maidenhead") from the flea, you can't lose your virginity from having sex. We'd guess that he probably doesn't think the woman is stupid enough to buy this ludicrous argument; he just wants to present himself as an awkward and bumbling charmer, like a Hugh Grant character in a romantic comedy.
This flea is you and I, and this Our marriage bed, and marriage temple is. (lines 12-13)
Marriage is the height of respectability in Donne's day, so the speaker tries to use the institution to his advantage. He playfully tries to convince the woman that their affair would be sanctioned – nay, expected – by the Church and even by God. They are practically making love inside a temple, for goodness's sake. It doesn't get more respectable than that, right...?
Yet thou triumph'st, and say'st that thou Find'st not thyself nor me the weaker now. (lines 23-24)
OK, so she doesn't buy the marriage argument. In fact, she literally squishes the whole little marriage-inside-the-flea. But the speaker comes right back at her, saying that if she doesn't feel weaker after having killed the flea, she won't feel weaker after having sex with him. It's another absurd argument, to be sure.
'Tis true; then learn how false fears be; Just so much honour, when thou yield'st to me, Will waste, as this flea's death took life from thee. (lines 25-27)
After a long digression, the poem comes full circle back to the idea that having a tryst would not be such a big deal. Once again, the speaker attempts to use the flea's small size to convince the woman that her "honour" (another word for reputation) would not suffer if they did the deed.