I doesn't modulate to c minor at all. It just cuts the scene and starts up in the new key. There IS, however, a pattern to the harmonic motion.
Throughout this song, the composer frequently moves to the relative minor of the tonic major of the current minor key. (Phew!)
Before your head explodes trying to unpack that last sentence, let me just show it:
Start of song
e-minor --> c# minor (which has same key signature as E major). This is a clear (and nice) modulation to c# minor.
This passage
e♭-minor --> c minor (which has the same key signature as E♭ major)
The difference is that in the first case, they modulated TO c# minor, but in this case, they are setting up a huge cadence to stay in e-flat (VI-vi-V-???), but do a sudden fakie to ( C# ) c minor anyway. This is NOT a modulation, in any meaningful sense of the word.
Short version: NOT a modulation, but still a planned part of this piece's harmonic progression.