Ok, so all cynicism aside for a bit, I can try to take a stab at the more direct question of "how to make something similar/functionally identical to the Plant SpikeBox"?
My take on it would be like this: take a programmable-gain amplifier (PGA) and buffer its output. This gives an analog signal that can be monitored directly with e.g. an oscilloscope or it can be ran through a analog-digital converter (ADC) and visualized through e.g. the Serial Plotter of the Arduino IDE.
The PGA should moreover be programmable through a microcontroller (that's where the Arduino comes in), and its output should be fed into an ADC (possibly built-in of the Arduino) coupled with the microcontroller. In the microcontroller's software an adjustment feedback loop can be made that monitors the PGA's output signal and adjusts its gain until a signal is obtained that manages to remain in-between the lower and upper extremes. Substantial and possibly adjustable filtering (analog and/or digital) will be needed to obtain a somewhat stable signal.
The actual sensor probe can be a simple wire held against the plant with a ground connection made with the growing substrate of the plant. See the Plant SpikeBox youtube videos for examples; this is really the easy part.
Getting an initial prototype would probably cost me (if I were to spend time on it) a day or two, not counting initial literature research to see how far I can get with what others have already published. Similar experiments and their methods have likely been published for many decades, so a good literature research will likely turn up some usable stuff. After the initial prototype it's likely that at least one other hardware iteration would be needed, as well as several software revisions.
Doing it right will not be a beginner's project. Getting some sort of signal and some flashing lights is fairly easy (see my earlier response), but meaningless. Getting it right of course also entails having a clear idea of what kind of plant responses would be measurable, what the sources of errors/disturbances are (this kind of setup is extremely sensitive to all kinds of false positives) and how to deal with those. Converting the entire experiment into a more durable setup that could for instance be put on display for a few days or weeks in a museum/exhibition space would be a challenge all of itself given the necessity for good physical contact between the sensing probe and the plant and the risks of losing contact, damaging the plant etc.
TL;DR: very easy to get some lights flashing if it doesn't have to mean anything, very challenging to make this work reliably if it needs to be somewhat meaningful (and that's not even touching the empathy part!)
PS: the PGA could either be constructed from a suitable opamp and for instance digital potmeter (and perhaps a DAC for a variable offset), or a suitable I2C/SPI-interfaced PGA could probably be found if budget allows it.
PPS: alternatively an AC detector is also possible, but it would require a programmable band pass filter. Otherwise the basic concept would remain similar.
