"exceed in any excess of evil," from Shakespeare's it out-Herods Herod in Hamlet's instruction to the players in "Hamlet" Act III, Scene II. Shakespeare used the same construction elsewhere ("All's Well that Ends Well" has out-villain'd villany). The phrase reflects the image of Herod as stock braggart and bully in old religious drama. The form of the phrase was widely imitated 19c. and extended to any excessive behavior.

Oh, it offends me to the soul to hear a robustious periwig-pated fellow tear a passion to tatters, to very rags, to split the ears of the groundlings, who for the most part are capable of nothing but inexplicable dumb-shows and noise: I would have such a fellow whipped for o'erdoing Termagant; it out-herods Herod: pray you, avoid it. ["Hamlet"]

by 1766 as a Southern U.S. derogatory term for "one of an inferior class of white hill-dwellers in some of the southern United States" [Century Dictionary], probably an agent noun from crack (v.) in its sense "to boast" (as in not what it's cracked up to be).

Cracker as "a boaster, a braggart" is attested from mid-15c. ("Schakare, or craker, or booste maker: Jactator, philocompus," in Promptorium Parvulorum, an English-Latin dictionary); also see crack (n.). It also was a colloquial word for "a boast, a lie" (1620s). For sense development, compare Latin crepare "to rattle, crack, creak," with a secondary figurative sense of "boast of, prattle, make ado about." This also was the old explanation of the term:

I should explain to your Lordship what is meant by crackers; a name they have got from being great boasters; they are a lawless set of rascalls on the frontiers of Virginia, Maryland, the Carolinas and Georgia, who often change their places of abode. [letter from colonial officer Gavin Cochrane to the Earl of Dartmouth, June 27, 1766]

For an alternative, DARE compares corn-cracker "Kentuckian," also "poor, low-class white farmer of Georgia and North Carolina" (1835, U.S. Midwest colloquial).

The word was used especially of Georgians by 1808, though often extended to residents of northern Florida. Another name in mid-19c. use was sand-hiller "poor white in Georgia or South Carolina."

Not very essentially different is the condition of a class of people living in the pine-barrens nearest the coast [of South Carolina], as described to me by a rice-planter. They seldom have any meat, he said, except they steal hogs, which belong to the planters, or their negroes, and their chief diet is rice and milk. "They are small, gaunt, and cadaverous, and their skin is just the color of the sand-hills they live on. They are quite incapable of applying themselves steadily to any labor, and their habits are very much like those of the old Indians." [Frederick Law Olmsted, "A Journey in the Seaboard Slave States," 1856]