Headset microphone is noisy, but touching speaker wires removes the noise
This can be caused by ground loops or grounding problems in general. You are applying a new ground point with your fingers when you touch the wire. I dont know the exact noise nor do i know the "touch"-setup. But maybe this indication might help you solve the problem.

Edit: Clarification: A ground loop or a poorly grounded wire can be vulnarable to EMI an thus can caputure interferences from other devices or power cords. This could be what you hear