Interesting question! Having visited the previous Guinness Book of World Records "quietest place on earth" at Orfield Labs, I have a personal interest in the answer as well.
To start, there's an article on the Brüel & Kjær website, which describes the instruments and technique used.
The measurement method was specified by Guinness, and used a two-microphone coherent power measurement technique with two Type 4955 low-noise microphones.
It's worth immediately noting that the B&K Type 4955 mics have a broadband inherent noise of 6.5 dBA, far greater than the -20.3 dBA of the room. This is what makes your question an interesting one.
So how did they do it?
With two very precisely in-phase microphones pointed at each other, the pressure at both microphone surfaces is known simultaneously, which allows a basic description of the pressure gradient between the microphones.
Brüel & Kjær describes the calculation of intensity from this (pg. 10):
The pressure gradient signal must now be integrated to give the particle velocity. The estimate of particle velocity is made at a position in the acoustic centre of the probe, between the two microphones. The pressure is also approximated at this point by taking the average pressure of the two microphones. The pressure and particle velocity signals are then multiplied together and time averaging gives the intensity.
So far, we just have a description of the technique - and we can see it's still limited by the inherent noise of the mic.
What I suspect the researchers did was to apply signal processing techniques to their measurement akin to those described in a freely-available paper by Ellingsona et al 2015. (In fact, one of the coauthors is an employee of Brüel & Kjær and the measurement technique is described using the exact same microphone and data logger used in the Microsoft anechoic chamber tests.)
The dual microphone, cross-correlation technique is based upon the premise that signals of interest will be highly correlated between the two microphones, while noise components will be independent and uncorrelated.
Note that this technique requires an unusually precise measurement of pressure change: ±0.05 dB. Most ANSI Type 0 (laboratory standard reference) sound level meters have a tolerance of ±0.40 dB, which explains why they didn't just use an "out of the box" type intensity probe setup.