"That's thirty for today" references an old expression meaning the end of a story. 19th century telegraph operators used "XXX" as an end-of-message flag to confirm that the message did not get cut off in the middle. Roman numerals XXX equal 30, so "30" came to be slang for the end of a story and, later on, the end of anything, including a speech, a work shift or even life.
The interviewer is meant to be Louella Parsons, a popular gossip columnist at the time. Bugs even calls her by her nickname "Lolly".
The title is a play on "A Tree Grows in Brooklyn," a coming-of-age novel set in early twentieth century Brooklyn. Not only is the book used as a sight gag here, Bugs actually is seen reading it at the end of the short.
The sign "Ice" on the door Bugs dances past indicates the household needed ice from the passing iceman. This was in the decades before the first refrigerators were invented.
As befits his Hollywood celebrity status, Bugs keeps his carrots in a monogrammed gold cigarette case.