I am sorry to say that I still do not understand what they did or why. I have no idea how many cells they tested or whether they even knew how many cells they were testing. The introduction does not inspire confidence. I really would like to think this study is meaningful but so far I can get no feel for what these lines mean.
Not really because I cannot work out the geometric relation between probe and a cell? several cells? - how do we know what the impedance is being measured across?
I would also like to see the raw renormalised impedances to see how variable they were.
"What" is a lot easier to answer than "why".
They produced "simplified blood" of each patient made up of PBMCs resuspended in plasma, 200 cells per microlitre.
A sample of this was added to the Nanoneedle chip and they then measured in impedance. Impedance is defined as the ratio of applied voltage to the induced current (if that helps).
Each chip has 4000 Nanoneedles, I think, and they used a sample frequency of five times a second for approximately three hours.
Once they had a study baseline measure they increased plasma sodium chloride concentration to 200 mmol per litre, to apply osmotic stress/increase energy demand.
Here are the diagrams from the paper showing the Nanoneedle design, and the captions:

(B ) Circuit model of a sensor−solution interface, where Zm-s is media−sensor surface interactions, Zc-c is cell−cell interactions, Zc-s is cell−sensor surface adhesion, Zc is a cell impedance (membrane capacitance Cm, and cytoplasm conductivity of the cells, σcp), and Rs is resistance of the solution. (E and F) SEM images of a nanoelectronic sensor tips, (E) top view
The "why?" Is more complex.
The paper states that "the array directly measures the impedance modulation results from cellular and/or molecular interactions". However, they don't offer any direct evidence of that, though more later on what they think is going on.
Interestingly, they weren't initially trying to develop a diagnostic test. Instead they were trying to set up a cellular model of post exertional malaise, using a high salt environment to ramp up ATP consumption and presumably lead to its depletion. Salt stress had been used to do that in other biological systems.
However the differences between patients and controls were so dramatic that they switched to investigating the Nanoneedle system as a diagnostic test.
They say the exact mechanisms behind the differences remains unclear and is they have a programme of work to try to narrow things down. They suggested several possibilities, including changes involving the endoplasmic reticulum and plasma membrane, and even increased cytokine production in response to the salt stress.
One possibility was change in size/shape in response to the osmotic pressure, but live microscopy at several critical stages in the experiment revealed no visible differences between patient and control cells.
Does that help? I don't understand the details of the different types of impedance measured.