There are two ways to connect the interface to the Buffalo DAC:
1. Synchronous clocking. This method allows one to synchronously clock the ESS chip, the advantage of doing so disables the asynchronous sample rate converter and DPLL in the ESS chip. In my opinion, running the ESS chip in synchronous mode sounds more natural, probably because you do not have the artifacts resulting from asynchronous re-sampling of the data. To run in synchronous mode you connect all four connections to the Buffalo: master clock, bit clock, word clock, and data, and then you remove the Trident regulator which provides power to the clock onboard the Buffalo DAC (you are now using the master clock from the interface instead).
2. Asynchronous clocking. If you want to run the ESS chip in asynchronous mode, just connect bit clock, word clock, and data lines, and leave the Trident regulator in place on the Buffalo. In synchronous mode the DAC is using the onboard master clock, and asynchronously re-sampliong the data.
Ground is connected via the U.FL cables, there is no need to connect a ground anywhere else to the interface (of course the power supply also has a ground connection). Note that the interface ground is also connected to chassis (if you use a metal standoff) through a resistor and a capacitor at one of the standoff mounting positions.