I put a lot of time into thinking and writing the safety page in the guide. I was hoping that with the schematic and common sense would help explain to anyone how to use and how not to use the device.
Basically if you are familiar how to use a scope safely, its in some ways similar. You can blow it up or damage it if you just connect grounds randomly.
The CR OUTPUT is not ground referenced (called GND_ISO in the schematic). That's why you cannot share that with the schematic Ground of the CR, or hook up multiple things to that ground, and especially the input ground which is at a completely different potential. In the best case doing things like that can give you invalid results.
The main idea is to avoid ground loops and unexpected current drains through the CR.
A bit of opamp knowledge and understanding of how CR is biased will help tremendously in avoiding damage by unintentional incorrect wiring.
The primary intent of the device is to measure current. If you apply a voltage differential, it will try to amplify it 100x within the 3.3v limits (of 3.3v/2 in bias/AC mode). That's how a load is able to generate a small drop across the shunts internally.
It's always OK to connect the output to a scope. Connecting other things to the input or anywhere in the circuit might not be OK if they are referenced to the same ground as the scope. I would use floating instruments if you plan that.
Best idea is to keep everything floating WRT each other. CR is itself floating internally (battery powered). A DMM is also floating so it doesn't matter where you measure between 2 points on the CR at all. A floating scope is great but if that's not available then keep everything else battery powered and away from earth ground to avoid ground looping through the CR.
FWIW:
If its of some relief, I was never able to blow up a CR or any of the 5 prototypes before R1, and it took over 1 year to develop and test and measure endless tests. I am sure I hooked up things wrong many times, unintentionally. Sometimes the amps will drift badly then come back to normal after a while. Large temperature swings for instance are introducing dramatic offsets in the max4239 amps, but as they cool down they go back to normal. Even if you blow hot air you will see offsets (and they get amplified 100x!).
The guide attempts to explain the safety so that you don't put the CR through a lot of unnecessary stress by hooking up things causing ground/current loops, as you should not do to any instrument.