"The next day" is an expression that can be used either as a noun or an adverb. There are many such expressions. Other examples are: "today", "yesterday", "tomorrow", "the day before", "the year before (or after)", "the day after", "the previous day", "next year", "last year", "mondays,..." In your first sentence, "the next day" behaves as an adverb that modifies "happened". In your second sentence, "the next day" is a noun which is the object of the preposition "on". Here are some examples of nouns acting as the object of a preposition. In each case, if the preposition is removed, the noun changes into an adverb: "What happened on mondays?" "What happened before (after, until) yesterday" "What happened during last year?"

Both of them are perfectly correct. This is an example of ellipsis. From the point of view of grammar, you would think that the preposition "on" is necessary. But it is not wrong to leave it out, because it is perfectly clear to what it means. You could say that a native speaker mentally puts in "on" automatically and doesn't even know they are doing it. Because of this, "What happened on the next day" sounds very slightly more formal. On a written English exam, I would probably use the second form "just to be sure."