Very simply:-

Flash drives are designed to behave in the same way as an external hard disk drive and internally the controller addresses the flash as if it was a standard hard disk.The usb interface is simply a high speed serial interface so the pc sends a command to read or write a file from the flash over the usb interface, the internal controller converts the request into read/write commmands for the flash which fetches/stores the data which is sent over the usb interface and Bob's your uncle.
There is a lot more magic involved obviously but the thing which allows it to work efficiently is the design of the physical signalling which goes on over the interface and allows data to b sent at a fairly high bit rate without too much hassle, which is acheived by a differencial voltage driver over two wires which act as two halves of a signal thereby eliminating a lot of problems with cross talk and noise.