Wordorigins.org

To come off with flying colors is to achieve great success. It is a military metaphor for leaving the battlefield still in possession of one’s flag. The phrase dates to the 17th century. From John Locke’s 1692 A Letter Concerning Toleration:

It may...bring a Man off with flying Colours.

And from George Farquhar’s 1706-07 A Beaux Stratagem:

We came off with flying colours.

Some claim a nautical origin for this phrase, but it appears to have got its start among armies, not navies.

(Source: Oxford English Dictionary, 2nd Edition)