When thinking about the Oxygen Destroyer, it's best not to think too deeply. Like, if you get rid of all the oxygen in H2O, how do you have any water left? Wouldn't you just have a bunch of hydrogen gas? Oh, and how does the Oxygen Destroyer know when to stop destroying oxygen? How does it not murder every fish in every ocean? For that matter, how does removing the oxygen result in fishy skeletons? Wouldn't they just suffocate?
The point is that the Oxygen Destroy is scientific claptrap when scientific claptrap was all the rage in movies. While we're on the subject, there's no ancient civilization at the center of the earth, no Martians are chilling on Mars, and shrink rays would just be an excruciatingly painful, not to mention expensive, way to die. As a symbol, however, the Oxygen Destroyer works pretty well.
Dr. Serizawa accidently stumbled across the device while studying all things oxygen. He decides to not publish his research until he can find a way to use the technology for the unequivocal good of humanity. You may ask what's to keep another scientist from reverse engineering the tech to its weaponized form once he does publish, but remember that we're putting away our nitpicks for now (as hard as that may be).
The Oxygen Destroyer basically represents how science discoveries can be co-opted as a way to harm people, rather than benefit them. Serizawa spells it out for us when he tells Ogata why he hasn't used the device against Godzilla:
SERIZAWA: Ogata…if the Oxygen Destroyer is used even once, the politicians of the world won't stand idly by. They'll inevitably turn it into a weapon. A-bombs against A-bombs, H-bombs against H-bombs. As a scientist—no, as a human being—adding another terrifying weapon to humanity's arsenal is something I can't allow.
Putting a finer point on it, the Oxygen Destroyer symbolizes the next logical step in the evolution of nuclear weapons (in effect, something far more dangerous than our current arsenal).
You can see this with basically any weapon in military history: ingenuity and scientific discoveries find ways to make a deadly device or technology deadlier. Bows and arrows gave way to rifles, which gave way to machine guns and automatic rifles. Horses gave way to chariots, which gave way to automobiles and tanks. Bi-planes gave way to fighter jets, which gave way to stealth fighters and combat drones. And on and on and on.
But with the atomic bomb, we reached a point where one weapon could effectively wipe us out. And what did we do? We developed the deadlier hydrogen bomb. Of course, we didn't stop there. We found ways to put nuclear warheads on strategic ballistic missiles and developed tactical nuclear weapons like land mines, depth charges, and torpedoes.
What's next? Who knows? Given how farfetched nuclear weapons would have seemed before the 20th century, it's possible that something as seemingly ludicrous as the Oxygen Destroyer may be next in line.
Although we're doing a pretty good job of destroying the air even without it.
A Permanent Solution
As Serizawa says, if he publishes his research, he'll just be adding another terrifying weapon to humanity's arsenal, which will only be more terrifying as time goes on. His solution is tragic, but also ties the Oxygen Destroyer and its symbolism into the movie's message of pacifism.
Serizawa agrees to use the Oxygen Destroyer to defeat Godzilla, but before he does, he burns all of his notes and research "to make sure it doesn't fall into he wrong hands." During the confrontation with Godzilla, he chooses to activate the Oxygen Destroyer himself and commits suicide so his knowledge can't be used to develop another weapon. The message is clear: we must do all that we can to ensure that the technology we develop must benefit peace and prosperity.
Given how far Serizawa went to keep his conscience clean, it's too bad he didn't work for the ACME Company. He would have saved that poor coyote a world of hurt