Power outlet inside a kitchen island. How do I protect the back?
You need to mount an electrical box to the wall of the cabinet and put the outlet inside it.  

If you can't easily do the work inside the cabinet, you can use a thin surface mount box on the outside.  The box screws to the cabinet wall, and you drill a hole through the cabinet wall behind the box to feed the wire into the box.  A standard wall plate goes over the outlets.

Here are a couples of examples from Home Depot:

Non-metallic example: (this one happens to come complete with outlet and coverplate)



Metal example:



As DrMoishe Pippik points out, a kitchen outlet needs to be GFCI protected.  If the wire was pre-run, it should come from a GFCI source, perhaps one of the protected kitchen outlets.  You could check this by temporarily connecting a load, like a light, or a voltmeter, and tripping the kitchen GFCI outlets with the test button.  There are also inexpensive outlet testers that have a built-in GFCI test button.  If the line is protected somewhere else, the button will trip it (and then you can search for its location, like in the garage or basement).  If the wire is not protected, use a GFCI outlet on the cabinet.