After throwing away my fifth pair of $15 earbuds i was sick and tired of these things breaking, so when this pair broke i got out my X-Acto knife and started cutting.

Step 1: Materials

The problem spot is usually right at the end of the headphones (duh). The problem is that the wires get bent so much that they break(another duh), but this can be easily repairer with the following materials:
.wire stripper
.different sizes of shrink wrap
.soldier
.soldiering iron
.X-Acto knife
.copper sheet (optional)
.matches

Step 2: Start Hacking

using the X-Acto knife CARFULY remove the casing at the end of the headphones. Then, take the wire strippers and remove about 3/4 inch of the protective wire casing.(pic 2)

Step 3: Reconnecting the Wires

You see there is a location where the good wire ends and the bad wire begins, cut the wires there. Re-twist the frayed wires the best you can and separate the wires so there is three distinct colors. Now slide some small shrink wrap over the long end. Match the colors together and soldier them together one at a time.

Step 4: Testing

Pull out your ipod, mp3, or other music playing device and plug in your headphones. Move the wires around she what creates the best connection. Once you have sound coming out the headphones the way you would like slide the shrink wrap up to the end and heat it up using the match.

Step 5: Finishing Up

Siffen up the end by sliding somebigger shrink wrap over the whole fixture ant the end and heating it up. I ended up with three layers(including the one mentioned on the step before). This provides a stiff and protective coating that should prevent this problem from occurring again.