I don't know what there is to explain. A counter melody is a second melody that contrasts the first (when played simultaneously).

Playing multiple melodies simultaneously is called polyphony, or counterpoint.
Only play what you hear. If you don’t hear anything, don’t play anything.
-Chick Corea
Somebody wasn't listening...

In my restless dreams...
I see that town.
Silent Hill.
You promised you'd take me there again someday.
But you never did.

Well, I'm alone there now.
In our 'special place'...
Waiting for you.

Its a second melody that is subordinate to the first. It complements the the melody, however it is not a harmony. It is completely different but still related.

Look up counterpoint if you want to know more. Its cool stuff.