The story
From fetch + and + carry, originally a reference to a trained dog fetching and conveying an object back to its master: see, for example, William Shakespeare’s play The Two Gentlemen of Verona, Act III, scene i (spelling modernized): “She hath more qualities than a water-spaniel, […] She can fetch and carry: why a horse can do no more; nay, a horse cannot fetch, but only carry, therefore is she better than a jade.”
Etymology from Wiktionary, CC BY-SA 4.0.