I think it works via "experience".
Each crew member has two values - their experience and the value required to become eligible for the next rank (e.g. 207/250, meaning that person needs another 43 experience at least). They earn some experience each patrol, which may or may not be a function of each patrol's 'success'.
If you look at the officers at the highest rank, you'll see they have a current experience greater than the value for the next rank as they are at the maximum rank i.e. they can't get promoted, but keep gaining experience anyway.
Others with greater technical knowledge of how the mod might affect this can give the definitive statements on exactly how this works (if it differs from what I've said).
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