It's about a sexual deviant that endangers national security while pursuing her rather unusual fetish for a giant Fish Man.
Most of the film focuses on her and the rather uninteresting life that she leads, which consists of working as a janitor at a facility of some sort for the military where the Fish Man is being held. She appears to be quite lonesome. Her only real friend in the world is an unemployed artist that spends most of his spare time sexually harassing the staff of a local diner.
The Fish Man is apparently quite valuable, to the point where generals and other top brass are willing to micromanage every detail of the military's investigation regarding exactly what the Fish Man is, what his capacities are, and how said nature and capacities might be used to the benefit of the United States -- for, perhaps, the space program, they say. Or so they say. So far as the viewer can tell, most of the research consists of Michael Shannon physically abusing poor Fish Man with a baton and prod. I'm guessing he did not major in Biology when he was in school.
Unfortunately for them, this ultra-important investigation is not important enough to require the cleaning crew to submit to any kind of security clearances, which, by the story's end, I'm guessing they would be willing to consider an oversight. Similarly, they don't bother to check the work history of the top scientist on the project -- or his references, or his background, or his educational history. Again, another oversight, surely, but it is worth noting that people at Burger King are screened more thoroughly than the personnel on this project.
The main character develops a bond with the Fish Man after mopping the floors one day. Said bonding experience basically just consists of her seeing him through some glass, making a little eye contact, and then feeding him a small snack, all of which is roughly on par with the everyday interactions at your local pet shop (no, she didn't tap on the glass while going "oh, cute!"). After this hypothetically meaningful experience, she decides that she just has to rescue Fish Man from the military, because the military is mistreating him, because there wouldn't be a story if everyone in the film was nice.
The movie limps along on the premise that if you have a problem with the idea of her being in love with Fish Man, then you are a bigot. By the same way we shouldn't judge people for their skin color, or their sexual orientation (even if said orientation manifests itself as a creepy old man sexually harassing diner staff? but I digress), then we should not judge her and her Fish Man. Not the most compelling argument I have ever heard. I don't see how it is an incoherent position to hold that while being Gay is certainly okay, falling in love with a giant fish is a bit questionable. But hey, that's just me.
Or perhaps it shouldn't be taken as anything other than a touching love story? Well, then you have a formulaic, by the books romance for simpletons. It wasn't touching or unique, it was just lame. I didn't care about any of the barely-developed characters or the contrived plot mechanisms that carried them through the story. There was no tension to any moment of the story, and the payoff was about as meaningful as your typical car insurance commercial.