Recently, I've seen fares on trenitalia listed as CarteFreccia Special. Can somebody explain how these fares work / who can buy these?
It's a frequent traveler program and you must subscribe, but the rub is you have t have a residence in Italy in order to subscribe. There are specials for everyone, though. There are advanced-purchase options (Economy, super-economy) as well as "Offers" (see the drop down on the English language site) such as 2 for 1, same day return. If you have children traveling with you, note the "Bimbi Gratis" fares are a terrific option.
Bob, first thing to do is click on the British flag icon to translate the page into English, you'll see some very good offers, especially for same-day roundtrips or weekend specials.