What does it mean by "Revenge is a dish that tastes best when served cold"?

What it means is that revenge is often a hot blooded response to a hurt, a furious lashing out as a reaction to the hurt. However, if you want to truly punish your enemy you must stop, and, in a cold and calculating manner, determine what and how to return and recompense yourself for the hurt he has done you.

The food metaphor is fairly random, the basic meaning is cold blooded revenge is better than hot blooded revenge. Puzo did not originate the expression, it has a history before him. However, it is associated with him because in some people's mind it is a good characterization of one of the core sub-themes of "The Godfather", and particularly the philosophy of Michael Corleone.

Whether that is true, or whether revenge actually is best served cold, I will leave as an exercise for the reader.

Wikipedia discusses "revenge is a dish best served cold":

The proverb suggests that revenge is more satisfying as a considered response enacted when unexpected, or long feared, inverting the more traditional revulsion toward 'cold-blooded' violence. In early literature it is used, usually, to persuade another to forestall vengeance until wisdom can reassert itself. This sense is lost in recent presentations.

Strangely, this is not what my initial reaction to the phrase has been. Having first heard it spoken by Klingons I had assumed it was a pun on dying in space which — as you know — is rather cold.