The metal inside the outer sheath is the conductor in this cable. and the inner core is just a former around which the conductor is wrapped.
So you have to cut through the outer sheath without cutting the inner braid: if you go too far you just get those little silver-plated filaments you have encountered. The inner forming core is designed to be crushable when you insert the stripped wire into the speaker terminal.
Consulting Editor, What Hi-Fi? Sound and Vision / whathifi.com
Audio Editor, Gramophone