Why do the Cirrus SR-20 and SR-22 have the CAPS (parachute) system?
Briefly, without the CAPS the aircraft could not be certified because it wouldn't meet the spin recovery requirements of 14 CFR 23.221.

You can read a lot of detail in Cirrus's own CAPS Guide but their design premise is that pilots are bad at recovering from spins, especially close to ground. Cirrus wanted to make a safer aircraft that could recover even if the pilot has no spin training at all, so they implemented two safety features: a "cuffed wing design" from NASA and the CAPS.

Normally, the Cirrus would have to comply with 14 CFR 23.221(a):


  Normal category airplanes. A single-engine, normal category airplane must be able to recover from a one-turn spin or a three-second spin,
  whichever takes longer, in not more than one additional turn after
  initiation of the first control action for recovery, or demonstrate
  compliance with the optional spin resistant requirements of this
  section.


But, rather than comply directly by demonstrating the spin recovery, Cirrus asked the FAA to certify them based on the wing design and CAPS instead, as the CAPS Guide explains:


  Given that Cirrus had demonstrated enhanced low speed handling
  characteristics that will help pilots to  avoid inadvertent spin entry
  and the presence of CAPS, the FAA granted Cirrus an Equivalent Level
  of  Safety (ELOS) for the spin recovery requirement of the
  certification regulations. This ELOS is accepted  by all civil
  aviation authorities that have certified the Cirrus SR20 and SR22


The CAPS also has some other benefits, such as being easy to use for an untrained passenger (think pilot incapacitation). That might be an important "why" reason and selling point from a Marketing perspective but it isn't directly relevant for certification.