Bdim is the VII chord - rather we should ask why is the Bb there! Bb is not in the key of C major!
However, great questions overall for a 'newbie' to music theory!
My guess... iii-vi-ii-V-I-VI-vii is a circle progression - iii-vi-ii-V-I-VI-VII-vii is a circle progression with a borrowed VII from the C minor scale (see below)
the vi–ii–V–I progression is a chord progression (also called the circle progression for the circle of fifths, along which it travels). It is "undoubtedly the most common and the strongest of all harmonic progressions" and consists of "adjacent roots in ascending fourth or descending fifth relationship"
- from circle progression wikipedia article
And also from the circle progression wikipedia article :
'The circle progression may also contain dominant seventh chords."
- Just don't know why they went with a flat! maybe they figure a key change around the seventh chord wouldn't be to drastic and allow some playful dissonance
Try playing that Bb, you'll notice its sounds slightly brighter than all the other chords, has a different mood. That's because its out of key, but just slightly, that it is still usable - its a borrowed chord (wikipedia has article on this too) - and there are not hard and fast rules for what you can and cant play! In music theory, this is known as a non-diatonic chord.
