Usually upon plugging in for the first time a USB drive in Windows a "Installing Device" dialog pops up to inform you that the needed drivers to use that device are being installed. Under Linux you need to have the right kernel modules for hardware support.
How are you able to use USB drives under UEFI and BIOS as bootable media? Do they carry the drivers or is there something special happening to allow them to correctly interface with the drives?
The main purpose of UEFI is to boot into upper-level OS. The UEFI is an interface between a platform-dependent hardware/firmware and a platform-independent (as much as possible) operating system. The interface does not dictate any particular implementation details. This initial firmware still can be called as BIOS, except that it provides higher-level functions such as boot and shutdown, instead of providing elementary "Basic I/O" functions as the original BIOS architecture was meant to provide.
And yes, the UEFI/BIOS does have its own blocks of code that are also called "drivers". Basic elements are embedded into UEFI code and loaded during Platform Initialization. UEFI starts with "Boot Manager", which looks at available (plugged) devices. However, to be able to boot into OS, the USB drive must be formatted in a certain way, and contain a piece of OS-specific software called "OS bootloader", and maybe other SW components. So yes, the bootable USB drive must be specially formatted to able to boot.
In essence, the UEFI in a modern PC is a specialized full-blown operating system by itself. For details, you need to consult with a series of published books on UEFI topic like "Beyond BIOS: Developing...".
