There are probably as many answers to this question as there are
April Fools.

The best one I have heard is that it goes back to the introduction
of the Gregorian Calendar in 16th century France. At that time,
France celebrated New Years Day on April 1st. When Pope Gregory
introduced his new calendar, he elected to move New Year’s Day to
Jan. 1.

The only problem was that the word did not get out with any great
speed about the swap of the holiday date. The people who continued
to celebrate New Year’s Day on April 1st became the first April
Fools.
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My mother told me that my grandfather changed his birthday from
April 1st to April 2nd after he came to this country to avoid being
called an April Fool. When I searched the family records in the
old country, I discovered his birthday was actually April 13th! I
wonder which bad date he was trying to avoid?