This is done using reasonably high-end fx software which effectively requires someone to go through the shot frame by frame and "paint out" the wires. Some of this software provides tools for automatating the process to some degree, but you still need to be a skilled image-manipulator. You certainly can't do this with Premiere... maybe After FX.

filmmaking.net

After FX production bundle can do it. Capture a still shot of the background w/out the wire and place it under the footage file. Then in the paint part erase the cable frame by frame and you will no longer have a wire. This only works if the camera does not move too much otherwise it would be quite trickey.

What? $1000!! That's crazy talk!!