Comment by metaxy2

To flesh this out, the root notes of C and A minor are not that close in the circle of fifths, but the chords sound similar because they share 2 notes (C is spelled C, E, G, and A minor is spelled A, C, E). A common device in major keys is to take a part played over the "1" chord (C in this case) and play it again, or some close variation, over the chord 2 steps down (A minor in this case) to get a "kind of the same, but sadder" version of the same part.