How do precision differential levels achieve such high accuracy
Obviously you don't need instrumentation like that to make granite surface plates extremely flat (they were made long before such electronics existed). You need three plates and a great deal of patience. See Foundations of Mechanical Accuracy on creating a master flat plane.
I don't know what's in that particular set of instrumentation, but one approach to detecting angles very close to vertical is a pendulum with differential capacitive sensor. It's not all that hard, with reasonable care, to detect fF of capacitive difference and ~10aF RMS is possible with extreme care over a 1Hz bandwidth. You can also make a closed-loop system using actuation to servo the gaps. Think excitation with kHz AC and synchronous demodulation, for starters.
I have an inexpensive mechanical master machinist level that has a resolution of 0.0002&quot;/10&quot; or about 0.005mm in 200mm. Smaller B grade (toolroom) plates are supposed to be about that good in overall flatness.