What is the advantage of the two-part rudder and how does it work?
This is called Split Rudder and it provides redundancy. They run on different systems so if one fails, the other one can be used.

Here is a picture of a split rudder:



Split rudders also provide a finer high speed control, in that only the lower one moves at high speed, reducing the exposed surface area and therefore the control effect. 

Very importantly, there is a structural benefit. When we use the rudder it imposes a twisting load on the vertical fin. By only using the lower rudder, when the aircraft is at high speed, we reduce the twisting moment and transfer that load to a bigger, stronger, part of the airframe. The principle is the same as why the outboard ailerons are disabled at high speed.

The patent information is here.