Here's the schematic, from the datasheet. Each stage holds the sampled voltage in a capacitor. The two clock phases push the capacitors up and down like a charge pump, dumping the charge into the next capacitor. Thus, the input signal appears at the output 256 clock cycles later.
This closeup of the die shows how the signal winds through the circuit. Each stage has two PMOS transistors and a capacitor. The horizontal metal lines provide the two clock phases and the gate voltage Vgg.