Here’s my take on it based on playing chords and scales over it and feeling the tonic. The written key signature (two flats) doesn't reflect the key changes.
If you want to question and validate this (you should), play the scales of those keys or modes (if you want to see the Fm6 as F dorian), and play cadences that lead to the claimed tonics. For example, to validate the claim that the F#m6 is a iv of C#m, press pause there and play F#m6 - G#7 - C#m.
For me, the Dbm7 - Fm jump has two possible “meanings” or interpretation: it also feels like a jump from iv of G#m a.k.a. Abm, which would mean a three-semitone change of tonic from Abm down to Fm.
The thing about such harmony jumps is that there are multiple plausible interpretations being “tracked” at the same time.
If you ask why these jumps "work", well - it seems that all the key changes (except when it wraps around from F#m to Gm) can be seen as three-semitone tonic jumps, which is common in jazz. But I'd challenge you to present a key change or chord combination that does not "work". Basically any combination of chords and/or keys makes you feel something. Whether you like that feeling, or whether you can create a melody that takes advantage of the feeling, is another question.
Anyway, thanks for the question, it's a nice tune I hadn't heard before, and it was fun to play over. Here's my polka reduction of the song.
Too bad the reduction didn't help you to feel the tonic jumps more strongly. :)