Neuronal membrane resting potential for large cells
In a typical neuron at rest, potassium is high inside the cell and low outside, with the opposite true for sodium. The membrane is mostly permeable to potassium. Let's ignore the other ions.

The resting potential in this situation will be something like -70 mV. Rest means that the net current flow is zero; however, there is still current: potassium is flowing out of the cell and sodium is flowing in. Therefore, if we turn off the sodium-potassium pump, over time, the concentration gradients will slowly equalize.

The authors are making an assumption that the current is roughly a function of membrane surface area: more membrane = more current. However, the "reservoir" of concentration imbalance is a function of volume: a bigger cell has more total potassium ions than a smaller cell.

Therefore, in a large cell, if the membrane potential is mostly driven by the concentration gradient, and it takes a long time for the concentration inside to equalize with the outside because the current is small relative to the volume, the membrane potential will only change slowly if you use a toxin to stop the sodium-potassium pump.

I don't think the authors are really intending to say anything special about large cells versus small ones, they are just setting up some assumptions under which their argument about the resting potential is going to be most evident. This is a bit like in a physics textbook where you read something like "assume a uniform spherical baseball."

The claim they are making: "the membrane potential is because of the concentration gradient, not the action of the pump itself" is true for all cellular membranes, not just in large cells. This particular piece of evidence they are referring to, however, is going to be most obvious in a large cell. That's all.

If you tried to do this experiment in a very small cell by turning off the pumps and watching the membrane potential, you might erroneously conclude that the pump was responsible for the membrane potential, but this would be a false conclusion caused by the concentration gradient equalizing too quickly. Instead you should test it in a big cell to see the effect more clearly.

In my personal opinion, this is not the best evidence to explain this feature, but it's a hard concept for many people to grasp and I suspect the authors of your text approach it in several different ways to find one that sticks.