This proverbial expression dates from the early 19th century, although versions of it that paraphrased the same thought existed well before then.
The first of these alternate versions is found in a biography of Marcus Aurelius by Jeremy Collier and André Dacier, titled Emperor Marcus Antoninus his conversation with himself, 1708:
You should consider that Imitation is the most acceptable part of Worship, and that the Gods had much rather Mankind should Resemble, than Flatter them.
A nearer stab at the current version comes in a piece by the English writer Eustace Budgell in the newspaper The Spectator No. 605, October 1714:
Imitation is a kind of artless Flattery.
The full monty as far as this proverb is concerned was given by Charles Caleb Colton, in Lacon: or, Many things in few words, 1820:
Imitation is the sincerest of flattery.
Colton was expressing the same idea as Budgell, in that, to imitate is to flatter without necessarily being aware one is flattering. As such, that 'artless' appreciation has to be 'sincere'.