It looks like "Tinsel wire", to use an old nickname...
https://en.wikipedia.org/wiki/Tinsel_wire
If it is that, one way to get it ready for soldering is to use a small gas flame, eg. a cigarette lighter.
One at a time, grip the end of a wire with something such as pliers or tweezers and move it across through the centre of the flame, about eg. half an inch from the end of the wire.
Get someone else to hold the lighter for you.
If you get it right, it burns away the plastic or fibre parts and leaves clean, bright metal, which you can then tin with flux cored solder.
Only trim the excess off after tinning it, otherwise it usually unravels and shreds itself...
To find out which wire is which, connect a 1.5V battery (eg. an AA cell) briefly across two wires and see which earpiece clicks.
If both do, you are between left and right. Use the other wire.
The "shared" wire, when you get a click with that and one or other of the remaining wires, is the ground. Left & right are then obvious.