There are a range of techniques. The most common are:
- damping using fretting hand: use the side of a fretting finger or an unused finger
- damping using the picking hand. If you are tapping, you will have spare fingers available, or the heel of your hand.
- wrap a damper around the neck and strings just above the nut.
I use all of those techniques. The damper is very useful for things like Joe Satriani's Midnight, but generally I use the other two techniques interchangeably, depending on what I'm playing.
Having watched the video a number of times now, I think the solution is partially damping using the edge of the tapping finger - if you watch closely he just touches the adjacent string as he taps - and partially down to the compression and distortion used on the effects: the new tap effectively hides the old note. I'm pretty certain he's not using fret hand muting (which you can do by slightly releasing pressure immediately before the new note)