Why not ask Higgins himself?
HIGGINS: "Garn"-I ask you, sir: what sort of word is that?
HIGGINS: It's "ow" and "garn" that keep her in her place,
Not her wretched clothes and dirty face.
Why can't the English teach their children how to speak?
This verbal class distinction, by now, should be antique.
If you spoke as she does, sir, instead of the way you do,
Why you might be selling flowers too.
Hooker, a Tolkien scholar, argues that "garn" is a "phonetic distortion" that should not make any sense:
It entirely misses the point that Garn! is a phonetic distortion that
is marked for the social stratum to which the speaker belongs. In
other words, it needs to be mispronounced and 'vulgar.' The context of
the dialogue in My Fair Lady at the point that Eliza says Garn! is
phonetics. Professor Higgins and Colonel Pickering are both
phoneticians, and they are discussing pronunciation. The ideal
translation of Garn!, therefore, should also be a phonetic distortion.
In other words, it is the phonetic distortion that is the most
important piece of information to convey in the translation of Garn!
in this context, not the exact semantic value.
Citing the OED, Hooker notes its origin as a pronunciation of "go on!":
It is not particularly easy to find a translation for garn. The Oxford
English Dictionary (OED) lists garn as an interjection, expressing
"disbelief or ridicule of a statement." It is marked as colloquial
usage, representing the--chiefly Cockney--pronunciation of go on! says
the OED. This explanation of its origin, however, belies its stylistic
marking. One of the examples in the OED indicates that its use is
vulgar, and this is the marking that is given in the usually thorough
Wildhagen German translating dictionary.4 The OED quotes the Glasgow
Herald from 1925: "He complained that if he used such words as 'garn'
or 'struth'5 he was accused of vulgarity ..."