Well, those statements need to be qualified because "automatic transmission" is no longer a well defined term now that there are a much larger number of different gearboxes on the market. So saying that "automatic gearboxes" are more fuel efficient is both true and false at the same time.

(For instance, one of my cars has a manual gearbox with an electrohydraulic actuator system and an automatic clutch, operated by flappy paddles. My insurance company says that it is an automatic, while the equivalent of the DMV (correctly) states that it is a manual.)

I drive a manual and I know exactly why it is not fuel efficient: the idiot behind the wheel :-)