The resistor develops a voltage drop that is dependent on load current. (I=U/R) Since reactive load, like loudspeaker, is not purely resistive but has impedance (IOW load resistance varies with frequency) the voltage drop across the "current sampling" resistor also varies with frequency.

This consequently varies magnitude of additional (negative) voltage feedback taken from that point. This effects closed loop voltage gain of the amp.

Dynamic loudspeaker has some archetypical impedance characteristics: Due to voice coil's inductance the impedance increases towards high frequencies. In addition, at resonant frequency of the speaker the speaker diaghram moves with greatest efficiency, which means the resonant frequency introduces a very high-impedance load in comparison to nominal load impedance. If nominal load impedance is, say, 8 or 4 ohms the impedance at resonant frequency and at upper high frequencies can be close to 100 ohms.

Google speaker impedance. Should be an eye opener.

You can now imagine a resistive voltage divider formed by the load impedance and current sampling resistor. When load impedance is moderately high (e.g. towards high frequencies and at resonant frequency) the resistive divider attenuates the additional feedback signal significantly and consequently voltage gain of the amp increases. When load impedance is low-ish (e.g. nominal impedance) the divider attenuates the additional voltage feedback signal less and closed loop gain decreases.

Thus the amp has higher voltage gain at higher load impedances. This shapes the frequency response: Gain is boosted at resonant frequency and towards higher frequencies.

Resonant frequency is typically near low end roll off of the driver while inductance increases towards upper high frequencies. Therefore result is boost of bass and treble, or in different terms: a mid-range notch.

If you think about an amplifier circuit with high output impedance (archetypal tube power amp) driving a reactive load it creates a similar frequency response due to similar divider formed by output Z and load Z. Simply, if output Z is high more voltage is developed across terminals of higher ohmic loads than across terminals of lower ohmic loads.

On the other hand, if output Z is very low (e.g. generic solid-state power amps) the divider has very little effect on voltage potential created across load terminals. This is great for high-fidelity since amp's frequency response is basically unaffected by load impedance. One acquires flat response despite impedance characteristics of the load. Very nice characteristic if additional "coloration" of the output signal is a no no.

With musical instrument amps (particularly guitar amps), however, we have grown to be accustomed to low and high frequency boost of a high output Z amp. So it's basically just another equalising circuit in the signal path. Amps without it sound lacking in bass and treble because they have flat response instead of boosting highs and lows.

This is a graph from whitepaper published by Fender's design engineers in early 1980's, which presents the difference of frequency response of high output Z amp (tube amp) and low output Z amp (solid-state) driving a reactive speaker load. In high output Z amp the impedance characteristics reflect in overall frequency response, in low Z amp the response remains moderately flat despite changes in load impedance.

Naturally, most SS guitar amp designers have been aware of this since late 1960's and have designed their SS amps to replicate the characteristic where frequency response is dependent on load impedance. For "Hi-Fi" (and largely for bass guitar amplification) such excessive coloration, however, is somewhat a "detrimental" characteristic. You mainly see this in guitar amps only.

So yes, it is true that this kind of circuit mimics similar behavior of tube (power) amps (but only those those that have high output Z) and likewise affects (closed loop) gain of the amp creating an impedance-dependent frequency response.