How does rubber toughening work?
I think this Wikipedia article answers your question quite directly:  


  "This important property [rubber toughening] is entirely due to the
  small rubbery domains that aid in arresting crack propagation, thereby
  increasing toughness/energy needed to break the material."


The same article gives an example using the materials sited in your post:  


  "Well-known rubber toughened polymers include high impact polystyrene
  (HIPS), rubber-toughened nylon, and acrylonitrile butadiene styrene
  (ABS). In HIPS, polybutadiene domains are dispersed in a polystyrene
  matrix; in rubber-toughened nylon, typically cis-polyisoprene is
  dispersed in a Nylon matrix; in ABS, polybutadiene domains are again
  used, but for both polystyrene and acrylonitrile."