What are specs of this MagSafe wire, and what is its purpose?
There are two cables that come out of the adapter (white brick) - a white cable and a bare cable (it's actually black on the very end of the cable at the adapter end).  The bare cable wraps around the white cable.  White being positive 18V and and black being ground.

Can you repair this?  

Yes.  However it will involve destroying the case of the adapter because it's held together with adhesive.  I did this once and had to reassemble it with duct tape.  However, instead of fixing the cable, replacements can be had relatively inexpensively on eBay.  

However, I highly suggest getting a new adapter because the case will never quite close properly and you run the risk of breaking the plastic because the adhesive is so strong.  If you do get a new adapter, get a genuine Apple and not a knock-off.