You can get the concept behind it (not the actual implementation) out of your basic trigonometric identities you may have learned in school. Lets say your input signal is a sinewave of a certain frequency: sin(w*t) , where w is your angular frequency 2*pi*f.

Now the first section is the 90 degree phase shift, as the article mentions, as it mentions this is the tricky party. But lets say you pull it off and you manage to get your phase shifted input signal sin(w*t-pi/2). The first rule we'll use is: a 90 degree shift in phase turns a cosine into a sine and vice versa, so sin(w*t-pi/2) = cos(w*t)

And then the magic happens, you multiply the cosine and sine input signal with a cosine and sine of your modulation signal, then sum them together

sin(w*t)*cos(m)+cos(w*t)*sin(w*t)

That equation remarkably similiar to the right side of first formula on this list here:

http://mathworld.wolfram.com/TrigonometricAdditionFormulas.html so the math says the output will be sin(w*t+m)

You use an oscillator for your modulation signal, and it basically ends up being added onto whatever frequency you put into the device