Yes, the maths can be a bit tedious sometimes.
In the simplest terms; imagine there is a little man controlling the fader of a single oscillator, he moves the fader in an up & down cycle (on, off, on ,off ad infinitum) as he increases the speed of the cycle eventually another audible frequency is generated.
Try it out yourself by turning the osc. balance hard left then use an LFO via the matrix to modulate the oscillator volume, then increase the speed of the LFO. Hear it?
That is basically how Ring Modulation works without having to trouble yourself with maths. Frequency Modulation is similar, but this time an LFO wouldn't be able to go fast enough so an audible oscillator is used to modulate another.
Here is an easy to understand guide to FM synthesis, start at the bottom & work your way up.
http://www.dubbhism.com/search/label/tao%20of%20fm
The "beating" you speak of, I'm not so sure. Let me play about a bit & get back to you.