Shouldn't these headphone wires be insulated?
They are insulated.  Each individual strand is coated in blue, red (is it red?) or clear enamel.

It makes soldering it a bit of a pain, but makes for very flexible wires, and also reduces the high frequency impedance of the wire by massively increasing the surface area of each wire.

It is called "Litz Wire", and the reasoning behind it is to reduce the Skin Effect which causes increased resistances in AC waveforms.