Here's the basic plot: a likable, romantic slacker* (Dean) falls in love with a pretty student (Cindy). (*He might be meant to be just working class, but he seems like a hipster, and when he applies for a job with a removal company he has no previous experience, coming across as a bit lazy rather than down on his luck.) When she gets pregnant by her stereotypically obnoxious jock boyfriend and can't go through with the abortion, Dean agrees to marry her. Presumably because of getting married and having a baby, Cindy goes from studying to become a doctor to being a nurse, and the family move to the country (near Scranton, PA, apparently) from Brooklyn, where they met. The marriage does not last, and the movie shows it falling apart.
Dean loves his wife and daughter (Frankie), and clearly wants the marriage to continue. Ending it is her idea, and by the end she cannot stand him touching her. So not much love there, and she sort of comes across as the bad guy, although he certainly has his faults too. Most obviously, he gets drunk and starts a literal fight at her work, which gets her fired. But he gets drunk because she repeatedly pushes him away when they're on a date at a motel and then leaves early in the morning without telling him where she is going (she is going to work, and she leaves a note, but it takes him a while to find it). The fight happens because he's drunk and because she refuses to speak to him. So the root of the couple's problem seems to be her changed attitude toward him. Why does she go from loving him to hating him?
Partly because he changes, but he doesn't change much. He goes from moving furniture to painting houses, from hip Brooklyn to the unlovely outskirts of Scranton, and from restless single to contentedly married. All he ever wanted, he discovers, was to be a father and a husband. It's a role he did not seek, but he settles into it happily. He's kind of a slob (but I wasn't sure whether his clothes were meant to be fashionably ironic or not), he smokes and drinks too much, and he is sometimes thoughtless (waking up Cindy when she needs to sleep as part of a game with their daughter, for instance). But she's thoughtless too, or careless, making oatmeal for Frankie badly and letting the dog escape into the road, where it is killed by a car.
It's not so much that he has changed as that their situation has changed. The motel they go to has themed rooms, and the one they end up in is called "the future room," or "the future" for short. It's sort of Star Trek-y and, as Cindy notes right away, it has no windows. So she is trapped in the future with her husband and can see no way out. This might be great if she loved him, but she doesn't. And that is apparently because he lacks ambition and is kind of immature. Being a man does not really appeal to him, which puts Cindy off.
The question of what it means to be a man is a big one in the film. Dean gets violent in the end in response to Cindy's demand that he be a man. What does that mean? A stereotypical man would smash things and hit people. Is that what Cindy wants? So Dean trashes the office and punches a doctor. But, of course, that isn't what she wants. She wants him to grow, to, in some sense, be all that he can be. But she can't really say what this would involve, other than developing his talents in some way. Settled contentment is not an option. Or not one that she can live with.
Another way that gender comes up is near the beginning when Dean says to a co-worker that men are more romantic than women because men marry when they find someone they feel they must marry, whereas women marry when they find a man who has a good job and is willing to take care of them. Dean is certainly willing to take care of Cindy, and he has a job, but he doesn't have a great job. She marries him partly because she feels the need for help in raising the baby she is going to have. Their love seems real enough at first, but she is pushed by other circumstances too. We also find out that she has had a lot of boyfriends (or sexual partners--she's answering questions at a clinic, so the terms are clinical) in her life, and started early. The nurse says this is normal, but we are clearly expected to be a little surprised, if not shocked. Is the idea that real life is not very romantic, or that she is not very romantic? I'm not sure.
If she isn't it might well be because of her home life. Her parents do not get along, which is illustrated by a scene (uncomfortable but a bit clichéd) in which her father refuses to eat the "disgusting" meal her mother has prepared for them. Dean's home life was unhappy, too, his mother leaving her father when he was young. He subsequently did not finish high school. So it's all a bit "man hands on misery to man." There is a sense of inevitable doom, that there is nothing to be done and no hope for anything better.
It would be possible to read the movie as very conservative. A kind of fatalism is traditionally associated with conservatism (hence no point in various progressive reforms--some people are just born bad, the poor are always with us, etc.--the opposite, pretty much, of "Yes we can!"). Dean could be regarded as doing exactly what feminists say men should do--being a nice, supportive guy--with disastrous results for him. And Cindy could be regarded as ungrateful, and her sexual history moralistically held against her.
But that certainly isn't the only way to see the film's story, and I don't think it is what is intended (apart from the sense of tragic determinism). Dean could have been just as supportive while doing something with his musical talents, for instance. And Cindy's relationships with men seem to be the result of her being so attractive (so men are always hitting on her) and her miserable childhood. She wants to be loved, and is perhaps desperate to be so. But that isn't all she wants. Is it that men and women want different things, or just that Dean and Cindy want different things? The film doesn't say.
It does seem to say, though, that love is not all you need, and that we are to a large extent slaves to our upbringing, our economic circumstances, etc. It emphasizes the value of individuality or freedom, the value of not being stifled and not dying (in the sense that giving yourself completely to others might seem to be a kind of death, as no doubt many housewives have found over the years). The couple's song begins with the words "You and me/ You and me/ Nobody baby but you and me." It's a good song, so it sounds good, but the lyrics might also sound dully repetitive and a little claustrophobic. Blue Valentine shows the clash between the romantic ideal of love and various less happy forces, as well as the ideal of autonomy.
Mostly, though, I think it suggests that this relationship was doomed. It's not so much "This Be the Verse" as another of Larkin's poems, "As Bad as a Mile":
Watching the shied core
Striking the basket, skidding across the floor,
Shows less and less of luck, and more and more
Of failure spreading back up the arm
Earlier and earlier, the unraised hand calm,
The apple unbitten in the palm.
But if you don't see life as being like this, then there is hope after all.