The regular expression:

(?-imsx:"(.*?)" "(((\")|[^"])*)")

matches as follows:

NODE                     EXPLANATION
----------------------------------------------------------------------
(?-imsx:                 group, but do not capture (case-sensitive)
                         (with ^ and $ matching normally) (with . not
                         matching \n) (matching whitespace and #
                         normally):
----------------------------------------------------------------------
  "                        '"'
----------------------------------------------------------------------
  (                        group and capture to \1:
----------------------------------------------------------------------
    .*?                      any character except \n (0 or more times
                             (matching the least amount possible))
----------------------------------------------------------------------
  )                        end of \1
----------------------------------------------------------------------
  " "                      '" "'
----------------------------------------------------------------------
  (                        group and capture to \2:
----------------------------------------------------------------------
    (                        group and capture to \3 (0 or more times
                             (matching the most amount possible)):
----------------------------------------------------------------------
      (                        group and capture to \4:
----------------------------------------------------------------------
        \"                       '"'
----------------------------------------------------------------------
      )                        end of \4
----------------------------------------------------------------------
     |                        OR
----------------------------------------------------------------------
      [^"]                     any character except: '"'
----------------------------------------------------------------------
    )*                       end of \3 (NOTE: because you're using a
                             quantifier on this capture, only the
                             LAST repetition of the captured pattern
                             will be stored in \3)
----------------------------------------------------------------------
  )                        end of \2
----------------------------------------------------------------------
  "                        '"'
----------------------------------------------------------------------
)                        end of grouping
----------------------------------------------------------------------