Evidently, the phrasing arose as a result of literal translations of certain phrases from Yiddish to English. Fred Kogos, From Shmear to Eternity: The Only Book of Yiddish You'll Ever Need (2006) identifies two such Yiddish phrases followed by their idiomatic and/or literal English meanings:
Zol ich azoy vissen fun tsores! I haven't got the faintest idea! (lit., I should so know from trouble as I know about this!)
Zolst nit vissen fun kain shlechts. You shouldn't know from bad (evil).
The form appears in Yiddish-inflected speech (and writing) such as this extract from Writers Forum, issues 21–23 (1995) [combined snippets]:
Then she [Mrs. Shapiro] sat down, took the biggest bowl of fruit which she topped off with a cap of prunes, popped an apple chunk in her mouth and said, "After you eat kichel [cookie], your kishkas [stomach or intestines] are filled up with air. That's no good. What you want to do is get your kishkas moving so now we eat fruit. It goes in and then it comes out. Believe me, you don't know from trouble with your kishkas." She shook her head in memory of some problem she had had with hers. "So, eat!"
And from this point of entry, the structure extends more broadly into non-Yiddish English. From J. Brent Bill & Beth A. Booram, Awaken Your Senses: Exercises for Exploring the Wonder of God (2011):
"I see people. ... They look like trees moving around." That's from one of my favorite Bible stories, in Mark 8. Jesus and his disciples come to the town of Bethsaida, and some friends bring a blind man to Jesus for healing. Jesus spits (!) on the man's eyes—a very human act. That shouldn't surprise us. After all, as Sara Miles says, "Jesus is the Word made flesh. While he lived among us, what he said and what he did were the same thing. His human body was God's language, as much as his human speech." So he spits and then speaks, asking, "Can you see anything now?" The man answers, "I see people. ... They look like trees walking around" (Mark 8:22–24).
I've always wondered how he knew it was people. And why he thought they looked like trees. How did he know know what trees looked like? I mean, he was blind; what's he know from trees?
Lilliam Feinsilver, The Taste of Yiddish (1980) has this entry for "know from" [combined snippets]:
"know from" for "know," "know of," "know about"
This influence from Yiddish is seen in various ways. First there is the extra "from" where English would normally use only 'know', as in the popular "He don't know from nothin'" and its derivatives discussed above. Among Jews, a similar usage shows up in the variations of "not to know from BORSHT' (IIIB).
As a substitute for "know of" or "know about," it is common in Jewish speech, as in the exclamation of a character in Jerome Charyn's fiction, recalling a poor childhood: "Who knew from school!" A line like this came through on TV in the Beverly Hillbillies in 1964, deprecating someone's acquaintanceship with English actors: "What does he know from English actors?" (Here the usual English meaning would be quite different.)
Also common among Jews are expressions like "You shouldn't know from such things" ...
FOLLOW-UP (January 8, 2015):
I just came across this discussion of "know from" in Leo Rosten, Hooray for Yiddish! (1982):
knows from ...
From the Yiddish: Vos vays ikh (zi, er) fun ...
The breezy substitution of "from" (fun) for "about" is a characteristic of Jewish New Yorkers' English in the 1920s and 20s. It is unacceptable. Such usage today is employed to characterize or parody:
"He's a Lehman, so what does he know from mama-loshn ["the mother's tongue" (i.e., Yiddish)]?"
"She's been married four times, so what does she know from loneliness?"
Rosten's comment strongly supports StoneyB's comment beneath the OP's question above that "know from" is an old New York Yiddishism that now functions primarily as "an ironic emphatic."