Could someone please explain the explanation given at the end of the page? The given explanation makes no sense to me.
It's probably easier to understand if written out differently.
We have $8x^3-2x^2-53x-39$ and $4x^3-3x^2-24x-9$.
To eliminate $x^3$ from the first equation we subtract twice the second equation from the first to obtain $$4x^2-5x-21.$$ We can now eliminate $x^3$ fro the second equation by subtracting $x$ times $4x^2-5x-21.$ This gives$$2x^2-3x-9.$$
You should be able to see these calculations in the table. The same process is now carried out with the two quadratics.