[octatonic]

All the chords are diatonic to C Major except the D.
In C Major the D would be a minor chord.

As far as 'why does this work'- there are a bunch of different ways of explaining how this works but that doesn't necessarily describe why it works.
That is a pandora's box of psychology, neurology, mathematics and the art of music.

If you want to look at the chords played using open chords you could observe some interesting things.

The Em would have an E on top.
The D would have an F♯ on top.
The F major would have an F.
The C would have a C on top.

The interesting bit (to me) is the semi-tone movement between the D and the F chords top note.
This is a rudimentary type of voice leading that you see a lot in jazz.