How do radios know the difference between noise and a signal?

A radio wave has a narrow band of frequency while noise has a wide band. Unless the amplitude of noise is too strong, your radio can lock on a radio station’s signal.

A radio does not know the difference. It is simply a circuit tuned to an RF resonance frequency and anything in the tuned range gets through. The signal to noise ratio must be such that there is more signal than noise. Sometimes noise can be filtered out, other times not. Most radios operate above the “noise floor” where the signal is easily distinguished from the noise by being much higher in power. Once you get near or below the noise floor forget distinguishing it.