Both my portable music player [FM radio mode] and my small AM/FM radio require me to insert the headphone cable to obtain signal reception.

I'm intrigued as to how this works.

The stereo headphone wires are presumably connected either to the device common supply rail or to an audio output channel. I'm not sure how an RF signal can then find its way to the receiver front end without significant or total loss along either of those pathways.

How did the designers achieve this function?

 

They likely just use a capacitor to couple the RF signal from one of the headphone wires to the radio input. They may add a small inductor in series with the headphone amp output to the wire to block the RF from being absorbed by that pathway. As far as the radio is concerned the cable then just looks like simple monopole wire antenna for the RF frequencies.

 

I sleep with earbuds and a portable radio. Something I've noticed is that one wire gets real hard to the touch over time.

Electrical signals must be making the wire fuse to the insulation, making them brittle. I go through two sets of buds per year.

Why is that?

 

Most likely due to absorption of skin oils or other external factors. The electrical signals are too small to have any effect.