White hairs inside a headphones cable
They are just essentially string to help support the cable. You should be fine soldering the cable.

One thing to remember about the cables in headphones is that it is usually enamel coated copper wire. You usually need to heat it up to around 390*C to burn off the insulation before you can make a decent solder joint. The way I do this is to put a large blob of solder onto the end of a soldering iron, then push the end of the wire that needs to be tinned through the solder and pull it back out again. This usually neatly removes the enamel and tins the wire on all sides making soldering easy.

Bonus info, when more than one of the wires is broken midway along the cable, the easiest thing to do is the cut each of the wires at staggered locations so that when you solder them you don't need to worry about insulating one wire from another as the solder joints will be at different locations and the enamel on the other wires will be intact on the other wires. If you are soldering on a new plug though, you don't need to worry about that.