A man named Eddie shows up to the motel room of a woman named May. The two apparently have a romantic history, but there's some pretty bad tension between the two of them now. Apparently, Eddie has run out on May before, and now she's torn between wanting him to stay and, it seems, not wanting to get hurt again.
Their conversation is already pretty tense, but then May mentions that she has a guy coming over. That really sends Eddie over the edge, and he storms out. Of course, because he (like May) is super volatile, he comes right back, suddenly convinced that May is lying—even if she's not, he intends to stick around to get a look at this guy (and perhaps beat him up).
May is less than thrilled.
While May and Eddie are talking, one of his former lovers shows up and blows out the windshield of his truck before driving off. May is livid, since she thinks the woman is "the Countess," someone Eddie cheated on her with in the past. In the aftermath of all that hoopla, May's date, Martin, finally arrives.
Instead of letting May going off on her date and bringing this whole weird toxic relationship conversation to a close, Eddie decides to entertain the new arrival with the story of how he and May got together… which is when we learn that they're half siblings, in addition to being former lovers. It seems they had gotten together back in high school, before they knew about their relation to each other.
Apparently, their father had been a bigamist, splitting his time between their two families. (By the way, said father is the Old Man, who has been at the front of the stage on a little platform since the beginning of the play; Eddie and May are the only ones who can hear/converse with him, though.) As they tell this story to Martin, they get increasingly googly-eyed over each other. Eventually, they start making out. Martin, baby, why are you still there?
After the incest bombshell drops, you'd think the surprises would be over, right? Nope. The Countess shows up again and basically blows up Eddie's car (apparently by shooting the gas tank). Eddie goes outside to check on the vehicle, and May tells Martin that he's totally not coming back (despite the fact that he said he'd be right back). So, she packs a bag and leaves, ostensibly to follow her half brotherlover.