The BiBiTel product is a decal with some embedded electronics. You remove your phone's SIM card, affix the decal, and put it back in your phone. The electronics in the decal passes through all communication used to place domestic calls. When you try to place an international call, or if you are out of the country, however, it intercepts the call and places it over the BiBiTel network instead of your normal carrier. The call is handled and billed by them at their discounted rates. (At least I assume they're discounted - I don't know what the rates are normally in the UK.)
Conceptually, it's not much different like land-line services in the past, where you would dial an access code or a toll-free number to place your call through an alternate carrier. Those methods (especially the toll-free number approach) would still work with a cell phone, but something like this is far more convenient. If it works as advertised, you no longer have to think about it - it will just redirect your calls automatically.
I never place international calls for myself (and work calls are on the company phone, which they pay for), but for someone who does, this is a great idea. I wonder when something similar will become available for customers of US carriers.