Well, it seems the exact reason is not quite certain. But majority of resources show that it is related to the reformation of calendar around 1582 in France. Before that Julian calendar was widely in use. Due to some astronomical reasons(which is worth another post in this blog) Julian calendar had to be reformed and that resulted into the adoption of Gregorian calendar which is the International civil calendar we use nowadays. The major change in this new system was the beginning of New Year on January 1, rather than on The New Year's Week, March 25- April 1, in the old system. But we can imagine in those days how fast news would have traveled. In fact it took several years for all to know about this change. Some rebellious fellows still continued to celebrate New Year on April 1 even after knowing about this change. These people were called 'Fools' by the general populace. They used to play pranks on them too. This harassment evolved over time and a custom of prank-playing continued on the first day of April. This slowly spread from France to Scotland and England and the found its way into America. Because this spread into other countries, April Fools' Day has taken an international flavour with each country celebrating the day in its own way.
And so it is, the Fools' Day, on which you are licensed to play prank on others.