Robert Hendrickson, The Facts on File Encyclopedia of Word and Phrase Origins (1997) offers a somewhat more elaborate version of the boardinghouse explanation:
applesauce. The expression applesauce for disguised flattery dates to the early 20th century and may derive from "the boarding house trick of serving plenty of this cheap comestible when richer fare is scanty," according to a magazine of the time. The term also came to mean lies and exaggerations. As a word for a sauce made from stewed, sweetened apples, applesauce is an Americanism dating back at least to the mid-18th century. Applesauce as a term for insincere flattery may also have been invented by American cartoonist Thomas Aloysius Dorgan (1879–1929), "Tad" having been the most proliifc coiner of his day. No one knows for sure.
Although Hendrickson doesn't identify the magazine in question, it is Century Magazine (Autumn 1929), in an article about Thomas Dorgan by Henry Robinson titled "Tad for Short: Cartoonist and Phrasemaker, a Victim of Circumstance." Here is a the full quotation from that source:
"Applesauce" means a camouflage of flattery, and is derived from the boarding house trick of serving plenty of this cheap comestible when richer fare is scanty.
As you can see, the primary definition of applesauce in this slang sense isn't "nonsense" but "deceptive flattery." But the "nonsense" meaning was certainly in widespread use in the 1920s. For example, from H.C. Witwer, "Auto Intoxication" in Collier's (January 1, 1921):
"Gimme my money back, you burglar!" shrieked the newcomer, ignoring me entirely. "I'm gonna have you pinched. This old tomato can is nothin' but a mess of junk! When I cranked it up last Friday mornin' the rear end fell out in the street, and the repair man tells me the only reason it ever run a foot is because you went to work and doped the motor with ether. The bearin's is all shot, it needs—"
"Ssh!" interrupted the dealer wearily, as one who hears an old story. "That's all apple sauce! I told you the job needed the touch of a monkey wrench here and there. You can't expect to git no factory pet for what you paid for this car. ..."
From H.C. Witwer (again), "The Shooting Stars," in Collier's (June 11, 1921):
"You were hitting it up so fast. I had no idea you intended turning off the road here," he goes on. "However, I'll—"
"That's all apple sauce!" I bellers, steppin' over to him." A guy drivin' a can like yours should be prepared for anything, and they ain't nobody in the world goin' to run me down and get away with it. Put up your hands, you big stiff!"
And from H.C. Witwer (again), "The Speaker Sex," in Cosmopolitan (November 1921):
In my widely ignored opinion, gently reader, this Otway baby [the poet Thomas Otway, 1652–1685] must of studied the speaker sex [women] at a correspondent's school or else he pulled the above roast right after some dizzy blonde had gave him the raspberry. The idea that women is poison is all applesauce. The Janes is the wine of life—the trouble is that they're a drink which few of us since Adam has been able to quaff like gentlemen!
J.E. Lighter, Random House Historical Dictionary of American Slang (1994) mentions a confirmed occurrence of applesauce used in the relevant sense by Thomas Dorgan in 1919, as unearthed by Leonard Zwilling in A TAD Lexicon many decades later:
They spill a lot of applesauce about big money.
But Lighter also notes a show-business sense of applesauce that appears a year earlier than that:
applesauce n. 1. Theat. silly, trite comedy. [First three cited occurrences:] 1918 Variety (Apr. 12) (vaudeville sec.) 7: Just to be back slipping on a little grease-paint, stepping forth with some comic to do "apple-sauce." 1923 N.Y. Times (Oct 7.) VIII 4: Apple Sauce: Hokum that falls soggy like a wet towel. 1924 Sat. Eve. Post (July 12) 15: Her routine's just a lotta apple sauce she's copped off other comics.
Lighter also notes instances where applesauce appears in a pair with another noun as an interjection "used in expressions of surprise or disbelief, with examples cited from 1884 ("Airthquakes and apple-sauce!") and 1889 ("Apple sass and spinage!")
Eric Partridge, A Dictionary of Slang and Unconventional English, fifth edition, points out yet another meaning of the term in British English:
apple-sauce. Impudence: mostly lower middle class: late C. 19–20. An elaboration of sauce, n. 1 ["Impudence, impertinence"].
Why 'applesauce'?
To understand why applesauce is associated with "nonsense," it might be helpful to consider what the three main definitions of applesauce as a slang term in the 1920s—"deceptive flattery," "nonsense," and "silly, trite comedy"—have in common. The shared element, it seems to me, is that the things characterized as applesauce are pitched to an unsophisticated, if not altogether gullible, audience. During the time period in question, I suspect, applesauce was broadly associated (in the United States) with rustic people and with children—two groups that many entertainers and confidence men considered rubes. As the Wikipedia article on apple sauce points out, "In the United States, packaged apple sauce is primarily branded as a children's snack, and is ubiquitous in school cafeterias." The idea of apple sauce as an inexpensive treat for children may go back to the days before it was widely available in packaged form.
Another possibility is that applesauce is a euphemism along the lines of horsefeathers (which Dorgan was again one of the first people to use in print) or bullcorn.
In any event, applesauce is not generally treated as a gastronome's delight. It tends to be bland and somewhat cloying, with a gruel-like texture, all of which make it suitable for relatively unsophisticated palates. It is (or was) also fairly cheap and easy to prepare in quantity, which may have inspired Robinson in 1929 to equate "the boarding house trick" of serving applesauce with resorting to "the camouflage of flattery." Nevertheless, it may be simpler and more accurate to think of applesauce as a plant-based alternative to baloney.