If you have a C chord, voiced C,G,E,G; and it moves to a series of pitches, voiced C,F#,D#,A. It moves back to the same voicing of C.
How does that use of pitches relate to the I chord? It creates a diminished chord where all the pitches are tendency tones to the I chord, while carrying the tonic through it. In C, a common tone diminished chord is spelled C, D#, F#, A. That’s a C (tonic) with a diminished chord on top (D#o).
Common-tone diminished chords have tonic function and serve to elongate the tonic function. It decorates a major I chord. These don’t exist in minor. Ideally, the I chord exists on either side of the common-tone diminished chord (I - cto - I), but that doesn’t always happen.