The short answer is that nobody really knows, but there are some interesting speculations. The first wide-spread mention of April Fools Day was in the 18th century and, even then, people wondered about this odd day. In 1708, a contributor to the British Apollo magazine wrote, 'Whence proceeds the custom of making April Fools?' It's hard to believe, but by this point, it was already a widely-accepted custom throughout parts of Europe.
Chaucer makes mention of it in his work "The Nun's Priest's Tales" written in 1392. He writes that a rooster was pranked by and, in turned, pranked a fox on 'Syn March bigan' thritty dayes and two...a clear reference to 32 days after the beginning of March...or April 1st. Scholars disagree with this theory and say that it's 32 days after March ends...or May 1st. The real reference to this day was made by Eduard de Dene, a Flemish poet, who writes about sending his servant out on fruitless, or fools, errands. In 1608, a French poem, Poisson d'Avril (April fish) tells about a "fishy" prank played on the unsuspecting, on the first of April each year. This had lead researchers to believe that April Fools originated in the northern European continent before traveling to Britain, and then the US. Even today. French school children will pin a drawing of a fish on the backs of a classmate. "Poisson d'Avril" is then yelled when the fish is discovered.
By the 1600's the legend of the Duke and Duchess of Lorraine's escape from prison had become popular. Story goes that they "fooled" the prison guards by simply walking out of the prison's front gate, dressed as peasants. It is known that the royal pair escaped in April, 1634, but it is not known if it was on April 1st. By the end of the 17th century, it became a popular prank to send gullable tourists to the Tower of London to see the "washing of the lions." Of course, no such event existed. Various examples of this hoax continued through the mid-1800's.
There are references to April Fools Day behavior, from the Roman celebration of Hillaria, celebrated on March 25th each year. The festival coincided with the spring equinox and the Romans were encouraged to participate in games, pranks, and masquerades. Other scholars believe that April Fools orginated from the Hindu festival of Holi and midieval Festival of Fools. Others, still, believe it had to do with a change of the calendar when the New Year's celebration was changed from January 1st to March 25th and lasted through April 1st. The French believe that poisson d'Avril stems from early fisherman who thought that the fish were easier, and more gullable, to catch in early April.
Although no one has, yet, come of up a definitive date for the begininning of this silly day, it is clear that it's been around a long time. You could choose to ignore this day or, you can join in the fun and come up with a few harmless pranks of your own. Like I always say, "If you can't beat 'em...join 'em.