Ok, it looks like the creator of that video doesn't really know either, though he makes some educated guesses. My thinking is that in Shakespearian plays, the fools were always secretly the wisest characters, and that April Fools really is just everyone being jealous of all the secretly wise fools that are born every April.
Makes perfect sense, right?
And, of course, I am in no way biased just because I was born in April. That's utterly foolish.
Happy Early April Fools Day!