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.

I believe the guide that you're after to repair your magsafe should be this:

https://www.ifixit.com/Guide/MacBook+Power+Cord+5-Pin+MagSafe+Connector+and+Cable+Replacement/21842

As to the detail spec of the shielding am not entirely sure.

Also, from the link I gave you, it seems like Amazon does not sell the replacement part any longer :( Might have to source somewhere else(At your own risk).

Alternative repair without replacing the cable would be this:

https://www.ifixit.com/Guide/Restoring+Apple+AC+Adapter+Broken+Cable/24487

Personally I would advise with all the hassle might as well head down to your official Apple store and get a new one. Official and safer. Just my opinion. :)

Hopefully those links will help you out :)

Cheers.