This is explained in the novelization (and, I believe, in the movie). When Grant arrives at the CMDF, he says that he was taught in school that miniaturization is not possible, for exactly the reasons you list: the only possibilities are to remove atoms or to compress matter, both of which are infeasible. He is then told that this is intentional disinformation, propagated to discourage investigation of miniaturization and to protect the secret that miniaturization is actually already in use. He's told that the process is analogous to photographic reduction: the atoms themselves are made smaller, they aren't removed or squeezed. This is complete hand-waving nonsense, but then at lot of real physics is counterintuitive as well.
One thing never explained is how they can see: relative to tiny eyeballs, visible light wavelengths would be huge and invisible. They'd have to use wavelengths somewhere in the upper X-ray range, which they wouldn't be able to see anyway, because their photoreceptors wouldn't respond to X-rays. The light waves must be getting miniaturized, too.
In fact, they must be inside a space warp. Yeah, that's the ticket: the Proteus just inhabits its own universe, which is smaller than our universe.
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