Does anyone know the origin of this term? And actually, does anyone know precisely what it means?

As I understand it, the idiom is used to describe someone who is (or at least believes themselves to be) clever, to the point of annoying others, usually due to outspokenness or arrogance.

I don’t have access to OED right now, so can’t look it up, but some googling suggests that the idiom “too [adjective] by half” has been in use for a good few centuries.

OED:

e. by half: by a great deal; much, considerably, far. Esp. in phr. too clever by half: trying too hard to be clever.
[...] ?a1400 Morte Arth. 2127 Thowe to hye arte by þe halfe, I hete þe in trouthe! [...] 1777 SHERIDAN Sch. Scand. IV. iii, Pshaw! he is too moral by half. 1858 G. J. WHYTE-MELVILLE Interpreter xli, Too clever by half. 1889 W. WESTALL Birch Dene II. vi. 89 ‘Nobody can deny as he’s clever.’ ‘Ay, too clever by half.’ [...] 1961 Listener 2 Nov. 717/1 A bad one [sc. documentary], whether dull through laziness, or self-conscious or pretentious or too-clever-by-half, can be a real catastrophe.

Nice one LH. Six hundred years is a pretty respectable pedigree for any idiom!

FWIW, two previous discussions of the phrase:

too clever by half

quite (sic) clever by half