Why is the Atlantic Hurricane Season defined as being between June 1 and November 30?
The hurricane season is set by the National Hurricane Center.  They use statistics to determine the dates.  Specifically:


  The Atlantic hurricane season is officially from 1 June to 30 November. There is nothing magical in these dates, and hurricanes have occurred outside of these six months, but these dates were selected to encompass over 97% of tropical activity.  June 1st has been the traditional start of the Atlantic hurricane season for decades. However, the end date has been slowly shifted outward, from October 31st to November 15th until its current date of November 30th
  
  NHC


As for why the dates are spreading out (from October to November), it appears that the dates of the hurricanes themselves are spreading out. (per James P. Kossin [1])

So, the dates are getting wider because the hurricanes are spreading out throughout the season.  So, the NHC widens the dates throughout the years to correctly cover the majority of the hurricanes.


[1] James P. Kossin - Is the North Atlantic hurricane season getting longer? GEOPHYSICAL RESEARCH LETTERS, VOL. 35, L23705, doi:10.1029/2008GL036012, 2008
James P. Kossin