Why is "hanged for this" correct but not "hung"?

Although this is more a matter of style than definition, the stylistic convention is that you always use hung unless you are describing a person (or I suppose animal) who has been killed by hanging—in which case it should be hanged.

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This is what Merriam-Webster says about it in "Is it 'Hung' or 'Hanged'?":

The standard rule for the past tense of hang is this: in almost all situations, you should use the word hung.

I hung a picture of Noah Webster on the wall.
After school, she hung out in the library.

Use hanged when referring to a person being suspended by a rope around the neck until dead.

The Salem "witches" were not burned; they were hanged.
Justice Wargrave ordered Edward Seton to be hanged by the neck until dead.

It's not that simple, however: most usage guides reserve hanged for people subjected to death, which means if an inanimate object is suspended from a gallows, the correct term is hung.

Despised by the voters, he was hung in effigy.

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To further illustrate this point, I wrote a short piece of fiction a while ago (I will emphasize the words):

The paintings of hanged corpses all hung in the gallery. As the exhibit was also a piece of performance art, the artist had put a noose around his neck and hung himself on the wall as well. Unfortunately, a spectator noticed that he was no longer breathing. When the police were called, they had no choice but to file a report about the man who’d both hung himself on purpose and hanged himself by mistake.