Why are aircraft registration details made public?
Aircraft registration are no more or less public than car registrations (although they are a bit easier to get access to). Generally speaking you can go down to the DMV and request data on a plate number, for example here is the form for doing it in PA (there is a fee involved). Since cars are registered by state its up to the states how this works, planes however are registered federally and thus governed a bit differently, the FAA provides a nice portal here for looking things up. 

From the FAA...


  In accordance with the underlying statutory framework (49 USC Chapter
  441) and as described in the applicable SORN, the information
  maintained in the aircraft record is available to the public upon
  request. This is to facilitate aviation safety, security, and
  commerce, the main purpose for which the record system exists. The
  public nature of the aircraft record allows for title searches by
  prospective buyers and/or financiers of aircraft purchases as well as
  for other interested parties to gather information concerning the
  aircraft including its airworthiness. The existence of recorded liens
  against the aircraft as well as the priority of those liens can be
  determined by a careful review of the documents maintained in the
  aircraft record.


This is done for a few reasons, first off many things in aviation may be billed by tail number. For example if you fly into an airport, say KPNE which has a city imposed 5$ landing fee, and the FBO for some reason does not collect this. The airport needs a way to look you up to send a bill. The FAA may also do this to make it easier for non-federal employees (local police etc) to access information related to an aircraft. 

For what its worth the FAA also lists pilot data pretty freely, however you can now elect to hide your personal data (but not your name)in the FAA portal.