Headphone wires often get lots of mechanical stress,
for example if you drop them, while the other end of the cable is fixed somewhere.
I think these hair-like things are fibres of some very strong material, probably a para-aramid like Kevelar, the fibres you make bullet proof and chinsaw proof clothing of - see below.
They are working as strain relief when otherwise the copper wire would get stretched and break pretty fast. The tensile strenght of kevlar is about 30 times the one of copper stranded wire.
DuPont offers Kevlar directly targeted at headphone cables:
From Kevlar® Ropes and Cables | DuPont USA:
For more than two decades, DuPont™ Kevlar® brand aramid fiber has
provided a lightweight, flexible, and dimensionally stable means of
helping to strengthen ropes and cables used across several industries,
from mountaineering ropes and fishing lines to electro-mechanical
cables and fine gauge cables for electronic device applications such
as mobile phone cables, computer power cords, USB cords, and MP3
earphone cables. Ropes and cables made of Kevlar® help deliver
performance and value to customers in the fine gauge cable industry by
providing excellent robustness, fatigue resistance, shrinkage, and
durability. The fiber’s resistance to chemicals and temperature
extremes makes it an ideal component for ropes and cables under severe
loads in harsh environments, from the bottom of the ocean to the
surface of Mars. Ropes made with Kevlar® brand fiber appear in a
variety of constructions, including 4-strand, wire lay, braided and
parallel yarns, and strand ropes. They range in diameter from 1mm to
250mm, and have been strength-rated up to 1500 tons.