To transform a failing object or venture into something that works effectively.

This phrase derives from the belief that bear cubs are born shapeless and have to be properly formed by their mother's licking. This belief is recorded in a translation of de Guilleville's The pylgremage of the sowle, 1413:

"Beres ben brought forthe al fowle and transformyd and after that by lyckynge of the fader and the moder they ben brought in to theyr kyndely shap."

[Bears are born foul and misshapen and are subsequently formed into their natural shape by the licking of their father and mother.]

The first use I can find of the figurative use of the phrase is in Gilbert Burnet's, An exposition of the thirty-nine articles of the church of England, 1699:

"Men did not know how to mould and frame it; but at last it was licked into shape."