What is this 5 second interval cut technique called?
It's referred to as a jump cut.  I would guess that perhaps the Apple ads that used it might have made it popular again, but that's just a guess.  The idea behind it is to visually show a change in direction of thought since typically it matches up with the start of another phrase.  It jumps the viewer forward in time and makes them aware of a subtle change in action.

A similar technique is often used in interviews where clips are cross dissolved to produce a smooth flow where silence is eliminated.  The approach of replacing these dissolves with jumps gives more of a feel of rapid movement through time and has made appearances in recent culture as more "edgy."