How does an oil tanker cargo such as Tanker Fame operates?

Answer 1

An oil tanker type cargo ship can carry either crude oil or products. Your question is a bit general, so first, the general answer:
The petroleum ("crude oil") is drilled for and delivered to the crude carriers. In Alaska the crude is carried from far inland by the Trans-Alaskan Pipeline to ports where the ships can load the crude as cargo. In other areas, like the Gulf of Mexico, the crude oil is far below the sea floor, and a production platform is used to collect the crude. The oil is sometimes sent ashore by pipeline; in other areas the crude is sent to a (floating) storage facility.
Crude carriers will load their cargo from this storage facility and take it to a refinery where the crude will be processed.
Many of the products the refinery makes are sent by ship to areas where the market demands them. This can be an established route (always taking the same product to the same place - like wax to the Crayola factory) or it can be a pot market situation (taking whatever product you pick up to wherever you can get the best price for it.)

Now the technical answer:
A tanker ship is typically a slower moving vessel than a passenger or container carrier. Tankers normally have either a slow speed diesel plant or a steam turbine propulsion system.