What does a USB or USB-C port actually do on a computer?
Think of it as road bridge. It joins the motherboard and the external device using a specific communication protocol. Almost exactly like those large slots on old motherboards. Its just really small now.

How does it access basically any and every function aspect that is available on the motherboard itself?
As a bridge each side has to know what is going to go over the bridge and what to expect on the other side. The data that comes into the motherboard is processed by a special chipset in charge of processing communication from external devices.

Camera, Audio, Hard drive and in this case ethernet connections?
Any device can be connected as long as it know the rules of how to send data across the bridge and the operating system has the "driver" to process the data correctly.