What are those weird wires in the audio cables?
Those are normal headphone wires.

The colored strands are copper insulated in a colored lacquer.
The core of each wire is a string (for strength as copper wire is not all that strong - especially not when it is as thin as this stuff.)
The combination of the very fine copper strands and the string makes for a very flexible wire that can stand being pulled on.

You can't strip the insulation off like you would with normal wire.
You would normally use a solder pot to strip and tin the wires all in one go.
A hobbyist won't have a solder pot handy, but you can still solder the wires.
Here's how:

Get a big blob of solder on the tip of your iron:



Poke the end of a wire into the solder:


The hot solder will melt the insulating lacquer and remove it.

Once the very tip of the wire has solder on it, push the wire through the solder from the side to strip and tin a couple of millimeters of wire:


It helps to have the solder standing by so that you can add solder as you are working with the wire.

You will end up with stripped and tinned wires that have a bit of burned lacquer at the transition from lacquer to tin:


Even the copper colored wire is insulated with lacquer - clear, in that case.

Strip and tin all of the wires, then solder them to the pins of the new plug:



As happens, I was working on a blog post on this very subject when your question came up.  I haven't published the post yet - I just finished making and preparing 37 photos for it and haven't written any of the text yet.

If you need a guide to replacing the plug on an Android headset, I've finished the blog post.  You can read it here.