Can I still use a damaged Magsafe power cable?
With the looming possibility that any advice you get from here could cause damage to your machine, I'll prefix my answer with a disclaimer that you should probably confirm with an official source or advice from Apple directly.

So long as the visible damage is limited to the insulation, you should be fine with electrical tape (as opposed to duct tape). The silver wire you see (likely braided) is used to prevent interference and is used as a ground.