These people are so clueless it would be hilarious, if it wasn't for the annoyance factor and the fact that they probably sucker in a lot of people. They have my old credit card information, and I've talked to guys with thick Indian accents who identify themselves with such names as "JR Downey" and "Jimmy Santos."
The scam is that you are automatically enrolled in some kind of subscriptions that you have to explicitly cancel if you don't want them. I didn't fall for it, but I'm guessing they're difficult to cancel and that they tend to lose track of the cancellation orders. They tell you this indirectly, buried in a pile of legalese delivered at lightning speed with an incomprehensible accent. They are clearly trained to take advantage of people who are easily confused or gullible.
As far as making them stop, my approach to these scams (which is working so far) is to repeatedly ask them if they are affiliated with Visa. Under pressure, they generally will. Then begin demanding contact information (street address, callback number, names, employee IDs, etc.) and let them know that you will be checking them out through the Visa fraud line before agreeing to anything. Don't get lazy, though...report the incident to Visa and, when applicable, report the telephone harassment to your phone company! Good luck.