With regard to the song in which you heard it, the phrase has a double meaning. That's probably why you have the impression that it means too much. The song is Silverchair's Ana's Song (Open Fire) and it's about anorexia (ana is teencode for anorexia). The double meaning is "to the bone" as in deeply (see @Kiamlaluno's answer) as well as in the sense of the skin-and-bone appearance of an anorexic body.
OED
The OED defines this bag as:
c. fig. A preoccupation, mode of behaviour or experience; a distinctive style or category; esp. a characteristic manner of playing jazz or similar music. Cf. bag of tricks at sense 18a. slang (orig. U.S.).
1960 J. Hendricks in D. Cerulli et al. Jazz Word (1962) 140 Lack of acceptance is a drag... Man, that's really in another bag.
1962 Jazz Jrnl. Mar. 30 ‘Bag’ is a current piece of trade jargon for hip musicians, and means something between a personal style and a body of work.
They further say:
(not) to be one's bag slang (orig. U.S.): (not) to match one's personal style, taste, or preference; (not) to form part of one's interest, preoccupation, or area of expertise. Usually in negative contexts. Cf. thing *n.*1 4d.
1966 N.Y. Times 20 Nov. d13/2 They were trying to categorize me..as a racial satirist, but that's not my bag. Let's say I deal in universal human foibles.
1961 examples
I found some 1961 examples of this sense of bag.
Billboard magazine (6 Nov 1961) contains the following in a list of new LPs to be released by Verve. Cal Tjader was a Latin jazz musician and Verve Records is an American jazz record label
STAN GETZ AND BOB BROOKME YER— V-V6-8418 (Nov.)
IN A LATIN BAG— Cal Tjader— V-V6-8419 (Nov.)
THE TRIO— Oscar Peterson, Ray Brown, Ed Thigpen— V-V6-8420 (Nov.)
BUDDY RICH BLUES-CARAVAN— V-V6- 8425 (Nov.)
Billboard of 20th November 1961 has an advert from Verve:
IN A LATIN BAG-Cal Tjader's torrid group in a program of crackling Latin-American jazz. Sensational sound!
And Billboard of the following week (27 Nov 1961) has a review:
IN A LATIN BAG Cal Tjader. Verve V 8419— Here's a warm and winning jazz set, which which combines Latin rhythms with jazz, and does it stylishly. Cal Tjader, with Armando Peroza, Paul Horn, Al McKibbon, Wilfredo Vicent, Johnny Re and Lonnie Hewitt, handle the charts with ease, and the disk marks a noteworthy debut for Tjader on the label. The tunes are mainly originals, sparked by by Tjader's "Davito" and "Paunetos Point," plus good readings of "Speak Low," and "Ben Hur," from the current flick. Lucid, meaningful jazz here.
A snippet of Down beat magazine dated 1961 includes a review of Tina Brooks' True Blue album, released in 1960:
Soul is appropriately earthy, medium tempoed, and melodically a bit doubtful as to what jazz bag it belongs in.
Best Answer
The OED gives the first citation of the phrase from Leon Uris, "Battle Cry" in 1953 (but if it appeared in print then, it would certainly have been around for a while before that. They also give a possibly related phrase "don't give a dead rat" from Mark Twain, "The Adventures of Huckleberry Finn" (1884).
COHA (Corpus of Historical American English) has its first instance in 1967.
If you're asking about how it came to be used, I think it is often futile to look for a particular story in the inventive ways that people use words - especially imprecations.