The technology Berger used in the 20’s was Electroencephalography or EEG. To perform an EEG a practitioner places 8-16 pairs of electrodes on specific areas of your cranium and measures electrical activity.
The difference in activity between one electrode and another is translated to a line which the practitioner can then use to determine what your brain is doing, and because the lines created by these machines adopt a wave like structure, the term brainwaves has been coined. It was thanks to the electrical nature of neuron to neuron communication that we were able to measure brain activity and it was because of this activity that we were able to start to study sleep in far more detail. So how is this measured?
Electrical waves like the example above are measured in two dimensions:
EEG machines measure the amplitude and frequency of electrical signals in the brain and transfer them to a printout sheet which looks like the below, and it is by studying these that researchers can reach conclusions about what is occurring in your brain during the night.
Note: Fig. 66 is used to give you an idea of how EEG readouts appear. Note that true EEG measurements would contain many more lines of data.