Rest assured that the fault lies not in your ignorance of maths and physics, but partly in the book itself, and more especially in modern physics in general.
The idea that energy has mass, should not be too difficult to grasp. Indeed it is clear that if sunlight did not have mass it would not actually exist; therefore as the plants on earth absorb sunlight, the earth itself becomes a tiny bit heavier. Similarly if you put a cannonball in a fire it will absorb heat energy, so the hot cannonball will weigh slightly more than the cold one did, although there are no scales on earth accurate enough to measure the tiny difference.
That gives us the equation e=m, not e=mc². To understand why physicists claim e=mc², check my review above.
If you managed to follow that, you will realise that the next stage is to understand why kinetic energy is defined as ½ mv². Probably this is explained somewhere in your book.
The final stage is to understand how the Lorentz equation is derived, which is not covered in the book, as it cannot be done with Einstein's special relativity.
That is why the book rabbits on about relativity, then at the crucial moment, the authors say they cannot actually explain the derivation as it is too difficult for the reader to understand. However what they really mean is that they cannot explain the derivation because they do not understand it themselves.
As Einstein put it "if you really understand something, then you can explain it to your grandmother". Much of Einstein's physics is more than a little dodgy, however his reflections on the human condition are almost invariably spot on.
Hopefully that will have left you a little less confused.