While I can't state the origin of the use of the specific term left as exercise, and further research is needed, I think it can be framed in a broader context.
Left as exercise can be considered a special case of proof by intimidation.
Proofs by intimidation are important rhetorical devices, useful when you don’t know exactly what you are saying, but you want to persuade the audience of your argument. They are studied in rhetorical logic:
[…] rhetorical logic, a promising field of science, of great value to
those writing research proposals. It provides new, and utterly
convincing tools for closing embarrassing gaps in your reasoning,
without resorting to brute-force methods.$^1$
The sharpest example of proof by intimidation is:
Proof: Trivial. $\Box$
Proofs by intimidation can be introduced by phrases as
"It is self-evident that..."
"It can be easily shown that..."
"Left as exercise".
Here a possible definition of proof by intimidation:
[...] “Proof by Intimidation.” The aim
here is to make something sound terribly difficult, using as much
jargon as possible, and then ending with “so obviously X holds.”
Though the argument may be completely obscure, even totally incorrect,
proof by intimidation is understood by everyone who is too vain to
admit they don’t understand you.$^2$
Left as exercise and Trivial are the most efficient proofs by intimidation, as they make immediately, and without waste of words, the audience feel stupid, and accept the thesis in silence.
The mathematician Giancarlo Rota said, in his book Indiscrete Thoughts, that the expression proof by intimidation was introduced by Marc Kac after the lessons of William Feller:
He took umbrage when someone interrupted his lecturing by pointing out
some glaring mistake. He became red in the face and raised his voice,
often to full shouting range. It was reported that on occasion he had
asked the objector to leave the classroom. The expression "proof by
intimidation" was coined after Feller's lectures (by Mark Kac). During
a Feller lecture, the hearer was made to feel privy to some wondrous
secret, one that often vanished by magic as he walked out of the
classroom at the end of the period. Like many great teachers, Feller
was a bit of a con man. — Rota, Gian-Carlo, 1932–1999. (1997),