[jmts]

AM reception is essentially the direct conversion of the strength (amplitude) of a given radio frequency into an audio signal. Any other noise present at the same frequency is added to the signal (superposition/interference) and therefore impacts the strength of that frequency at the receiver. Therefore it is impossible for the receiver to know whether the amplitude it received is just signal or is signal plus noise.

The claim 'the effect of random noise is to amplitude modulate' is probably not 100% correct, because to my understanding it's not actually performing modulation (the modulation happens at the transmitter but the noise happens between the transmitter and receiver), but it is impacting the amplitude at a given frequency and to a receiver this is impossible to know whether said change in amplitude happened before modulation (signal) or after modulation (noise).