Why descent below DA/DH or MDA could only be accomplished using natural vision?
The FAA states it pretty explicitly in their definitions: 


  Decision Altitude (DA). A specified altitude in the precision approach at which a missed approach must be initiated if the required
  visual reference to continue the approach has not been established...


and similar for MDA 


  Minimum Descent Altitude (MDA). The lowest altitude, expressed in feet
  above mean sea level, to which descent is authorized on final approach
  or during circle-to-land maneuvering in execution of a standard
  instrument approach procedure where no electronic glide slope is
  provided.


To over simplify it a bit: As noted in the comments, ignoring the not in use CAT IIIC approaches the whole point of an instrument approach is to get you down to a specified point where you should be able to see the runway. If you can't, then the weather has degraded and you must either try again or go to an alternate. The big idea being that at some point you need to be able to see the runway in order to land on it. Depending on how precise the equipment you are using is (ILS, GPS, RNAV, VOR, (AFD dare we say)) the height you can get down to without being able to see the runway changes. 

As for why there are minimums its likely historical, and at least in part practical. Not all approach systems/options take you right down to the threshold like say a circling approach and its possible to fly a poor approach that is within limits and not be directly in line with the runway when you pop out of the clouds. Thus you will need some time and distance to align visually, so minimums exist. On the historical side, early attempts at instrument landings systems we less than optimal to say the least and you would need time to pop out and align.