Surprisingly, this phrase doesn’t appear in the 1976 Supplement to OED 1. Perhaps someone with access to the online edition will be able to provide a more up-to-date answer; but on the assumption that the revision has not yet proceeded so far as the letter E, I offer this suggestion:
There is, first of all, a very old idiom—OED 1 s.v. Ear, with the earliest citation 1539:
1. d. † To go, come, fall together by the ears, be by the ears : said of animals fighting ; hence of persons, to be at variance (obs.). So To set (persons) by the ears : to put them at variance.
That’s not exactly self-explanatory; perhaps what was suggested originally was an image of animals (one early citation specifically mentions “dogges and cattes”) snapping at each other’s heads. At any rate, set by the ears endured for better than three hundred years. It began to fade in the middle of the 19th century, and it is very rare today: Google the phrase and except for dictionary entries you won’t find a hit less than a hundred years old until the end of the tenth page, in the English translation of Benedict XV’s 1921 Encyclical Sacra Propediem.
However, about the time set by the ears began to die a similar idiom arose in the US, listed in the 1976 Supplement, again s.v. Ear
1. h. to get (a person) up on his ears, to make him indignant; so to be on one’s ear; to get up or go off on one’s ear (U.S.), to rouse or bestir oneself.
1872 L. H. Bagg 4 Years at Yale 44 A man somewhat offended or indignant is said to be on his ear. [...] 1889 Farmer Americanisms s.v., to get up or go off on one’s ear, to bestir oneself; to rouse oneself to a great effort.
I think it's plausible to see a connection between set by the ears = “provoke a quarrel between” and get him up on his ears = “provoke him to quarrel” ⇨ get up on one’s [own] ear = “be moved to quarrel” ⇨ “be moved to exertion”.
Thus, in the 1930s there are two related idioms in the air:
“I’m sorry!” snaps the president of the Melvin High Dramatics Club, getting up on his ear. “But you’re talking to the wrong party, Mr. Moulton. I don’t bribe anybody with false promises.” —Boys’ Life, 1930.
Gammer Gurton’s needle is lost, and through the machinations of Diccon, the village ne’er-do-well, the entire parish is set by the ears. —Guide to Play Selection, National Council of Teachers of English, 1934.
And it is at this point that the new expression arises (despite the NGram, these are the earliest hits Google Books actually reports):
Years ago the scientific world was temporarily set on its ear by the discovery of numerous inscribed baked-clay tablets in Michigan. [...] The inscriptions resembled ancient Assyrian symbols. Proof of a lingual connection between the old world and the new would have been tremendously important. Still, archaeologists were unable to discover any traces of the tribe that had left these wonderful caskets.” —Popular Mechanics, 1933.
Hollywood was set on its ear when members of the Independent Theatre Owners Assn. [...] took a full-page advertisement in the Hollywood Reporter May 4 to declare themselves tired of losing money on pictures starring Joan Crawford, Greta Garbo, Mae West, Katherine Hepburn, Kay Francis, Marlene Dietrich, “whose dramatic ability is unquestioned but whose box-office draw is nil. Surprised were many movie-goers who had supposed that, in several instances, it was just the other way round.” —Life, 1938.
It looks to me like transitive set so-and-so by the ears was incestuously married to its intransitive descendant get up on one’s ear to create a new passive expression be set on its ear, meaning “be moved to controversy by a scandalously unexpected announcement”.