It's really nothing more than constructive and destructive cancellation when the carrier and signal become in and out of phase.
In it's simplest text book example with sine waves, the two frequencies are the carrier waveform and the modulation (signal) waveform.
If these were produced as sound and propagated independently, three frequencies are observed:
a) fundamental (hf carrier) = fc
b) information (audible frequency amplitude-modulation signal) = fm
c) the beat frequency of the two combined (harmonic frequency produced at carrier frequency +/- audible frequency) = |fc - fm|
The sidebands in AM are nothing more than the harmonics produced by the addition and subtraction of the two as they become in and out of phase.
The complex maths (Fourier) describes this beat phenomena as a trigonometric series expansion and allows modelling of complex waveforms by decomposing to an infinite series of summed cosine waveforms.