I don't understand even understand how to use the theorem, nor do I understand why it works.

Rational root theorem - Wikipedia, the free encyclopedia
In the example on the wikipedia page, it seems that they just took the 2 and 3 in the equation:

to get $\displaystyle \frac{2}{3}$, but where did the $\displaystyle \frac{1}{1}$ come from?

I was thinking about finding the point of inflection by the second derivative test to find a root. Is it true that the point of inflection of a cubic function will always be on the x-axis, and will thus give us the real root?