It's hard to say on a theoretical level why that minor-3rd relationship works, but you'll find that the minor-3rd modulation creates a bit of musical excitement that gives your song some momentum. The new key adds freshness to your music, and it allows you to easily extend the length of a verse.
What is a tertiary modulation?
Ah, well while studying Schubert at school, I was introduced to the term 'tertiary' modulation, which is a modulation to a key a major third up or down, but mainly used in the descending form, certainly by Schubert (ie C major to A flat major).
What are modulations in music?
Modulation, in music, the change from one key to another; also, the process by which this change is brought about. Modulation is a fundamental resource for variety in tonal music, particularly in larger forms. A short piece such as a song, hymn, or dance may remain in a single key.
How do you modulate tones?
Common Tone Modulation uses a sustained or repeated pitch from the old key as a bridge between it and the new key (common tone). The sustained or repeated note also belongs to the new chord, quite often with the aid of an enharmonic change.