This was one of the things that was most surprising to me doing software defined radio. It was "how does a finite impulse response filter work?" and "how to I make it work in a specific way?" The interesting question for synths would be how to give it an input that that changes its cut off frequency.

For me, the "aha" moment was when I connected the dots between "averaging" (which is a common way to filter noise out in an embedded computer) and "rolling average" is just summing the previous 'n' samples as sample * 1/n, and a picture of a fir filter that was (C code but it's pretty readable)

   out = 0;
   for (i = 0; i < n; i++) {
     out = out + sample[n-i]/n;
   }
   return(out);

That is a "FIR" filter where the coefficients are all 1/n. Now take that and do the FFT of n samples of n/1 (adding zeros to get the resolution you want). And that is the frequency response of your filter for frequencies between 0 (DC) and sample_rate/2.

For me at least that connected a lot of dots in what I was reading.