Stereo heaphone doesn't work properly when the balance is centered
Broken earth connection between headphones and plug. Earth connections on both phones are connected to each other, but not back to the amplifier 0V.

Balance fully right, the L channel outputs 0V and you get R channel sound from both speakers (they are simply connected in series, from active R, to L acting as 0V).

Balance fully left, ditto.

Balance centre ... both ends are driven with mono (L+R) signal, so both ends see the same voltage, so mono signal (vocals!) disappear. What you hear is L-R, i.e. only the stereo component.

Great way to turn any CD into karaoke...

Now you have to find out where the break is between the two copper coloured wires in the photo, and the sleeve on the plug, and repair it...

Don't worry about the 2 colours connected together. I can't see from the photo but it looks like the left cable goes to the plug, and the right cable just takes signal to the other earpiece, and the join you see is just a convenient way to make the connection.