In the context of a differentiator circuit, what is a “current-sensing resistor”?
What does the term “current-sensing” mean in this context?


When a current flows through a resistor, a proportional voltage difference appears between its ends, according to Ohm's law. Thus, a resistor converts a current to a voltage. If one end of the resistor is connected to GND, as here, then the voltage with respect to GND is proportional to the current flowing to/from GND through the resistor.

There are caveats, of course:


If the current is not from an ideal current source, but from a circuit whose behavior depends on the voltage across it, then the behavior of the circuit changes compared to if the resistor were just a wire (0 Ω, 0 V voltage difference).

In this particular case, the current is determined by the time derivative of the voltage across the capacitor. But by introducing the resistor, we've split the voltage \$V_{in}\$ between the capacitor and the resistor, so we've changed the current (see Chu's answer for the exact effect).
The resistor dissipates energy and gets hot. To minimize this effect (if the current is unchanged), you use a lower value resistor, but that means that the voltage is correspondingly smaller and more subject to noise. In particular, the resistor must be large compared to the resistance of the power supply wiring.
If you're measuring the voltage then that implies you're connecting it to some other circuit element, through which some amount of current also flows, disturbing the characteristic ("loading the circuit"). Hence, you want to connect \$V_{out}\$ to a high-impedance input like an operational amplifier or other buffer circuit.


Most of the above applies to any current-sensing resistor, not just ones that are part of a differentiator. Common uses of current-sensing resistors are in devices that measure current, like multimeters and energy usage meters. When they are large and low value, to handle large currents, they are often called "current shunts" instead.


  Why is the resistor even necessary?


As I mentioned above, decreasing the resistance decreases the voltage output proportionally. Imagine decreasing it all the way to zero; then the output is zero volts — the upper \$V_{out}\$ terminal is connected directly to ground, and you've turned circuit B into circuit A. So you've eliminated the resistor but also eliminated your output voltage.