Quote by Drakkith
That makes sense. However I think I'm failing to understand how changing the amplitude of the carrier creates two different frequencies in addition to the original two. It's just not "clicking" I guess.

It's not just that you're creating two different discrete frequencies.
The modulating signal (the one that contains the actual information content)
already contains a whole continuous
spectrum of frequencies. It's a theorem from Fourier analysis that you can represent an arbitrary timevariable signal as a sum* of sinusoids of different frequencies and amplitudes. So that information signal contains a continuum of frequencies, from 0 up to some maximum frequency (that defines the bandwidth). You would see this if you were to look at the frequency spectrum of the signal (which you do by taking its Fourier transform). Anyway, what multiplying this signal by the carrier does is simply to shift the frequency spectrum so that instead of one spectrum being centred on zero, there are now two identical such spectra centred on +carrier frequency and carrier frequency. To see why this is, you'd need to understand more about Fourier transforms and convolution.
*I use the term "sum" loosely, it's actually an integral called an inverse Fourier transform