My understanding is that a frazione is a hamlet. Can someome explain a little more. For example, San Costantino is a frasione of Briatico. Does that mean that there are different mayors, different town records, different Uffici Stato Civile? The wikipedia article said they do not keep their own civil records. Is this true?

Frazione is a part (portion, fragment) of something. A cent is a part, fraction of a dollar.
In the same way, Frazione is an administrative division of a municipality, it is a part of the comune. So a frazione does not have its own mayor, all the administrative stuff (including stato civile) is managed by the municipality to which it belongs.
In your case Briatico includes the following frazioni: Mandaradoni, Paradisoni, Potenzoni, San Costantino, San Leo, Sciconi, Conidoni.

Edited to fix a typo

Thanks for the fine explanation!