The proverb states a universal truth about human experience: a painful or unpleasant event makes you cautious when you encounter a similar situation again. An investor who lost money in a property crash may be once bitten, twice shy about real estate. Someone who was deceived in a relationship may approach new relationships much more carefully. The bite is the experience; the shyness is the resulting caution.
The phrase requires little explanation as a metaphor because the logic is so direct. An animal that bites you — a dog, a snake, an insect — teaches you to be wary of similar animals. You do not need to have the experience explained; it is encoded. The proverb merely formalises this instinct into language. The observation is ancient in concept, though the English form is relatively recent.
The English proverb is first recorded in Robert Browne's Jow Joe's Courtship in 1853. Earlier versions of the same concept appear in Chinese proverbs and European maxims. The phrase became more widely used through the 19th and 20th centuries and today is one of the most commonly cited proverbs in British and Australian English. It is sometimes expressed as 'once burned, twice shy' in American English.
It is an American variant of the same proverb. 'Burned' (as in burned by an experience) is a more common colloquial metaphor in American English; 'bitten' is preferred in British and Australian varieties. The meaning is identical.
As practical wisdom, yes — most of the time. But excessive caution after a single bad experience can prevent you from taking necessary risks. The proverb describes a natural response; it does not necessarily prescribe the optimal one.