A field $E$ is a simple extension of $F$ if $E$ = $F$($\alpha$) for some $\alpha \in E$.
What is this definition exactly saying? $F$($\alpha$) is a field of the form $p(x)$ = $a_0$$\alpha^0$ + $a_1$$\alpha^1$ + ... + $a_n$$\alpha^n$, where each $a_i$ is of the form $\frac{f_i}{g_i}$, and $f, g$ are polynomials with coefficients in $F$, right? What exactly is the definition of a simple extension saying in relation to this definition of $F$($\alpha$)? I'm not seeing the big picture here. Thank you.
Let $F\subseteq E $ be a field extension and let $\alpha\in E$. The we define $F(\alpha)$ to be the smallest subfield of $E$ containing both $F$ and $\alpha$. We say that $F\subseteq E$ is simple if we can find an $\alpha \in E$ such that the smallest subfield containing both $F$ and ${\alpha}$ is just $E$ itself.
Intuitively, $F(\alpha)$ is what we get when we add $\alpha$ to $F$, as well as all the extra elements required to make $F(\alpha)$ into a field (ie closed under multiplication, addition and taking inverses).
In practice we have two different options: we say $\alpha$ is algebraic if we can find some irreducible (monic) polynomial $p_{\alpha} \in F[t]$ such that $p_{\alpha}(\alpha) = 0$. This is called the minimal polynomial of $\alpha$, and we can show it is unique. Otherwise, we say $\alpha$ is transcendental.
Let's look at the case where $\alpha$ is algebraic first. Then if $p_{\alpha} = t^n + a_{n-1}t^{n-1} + \cdots + a_0$, we know that ${\alpha}^n + a_{n-1}{\alpha}^{n-1} + \cdots + a_0 = 0$. In particular, if we let
$$\beta = -\frac{1}{a_0}(a_1 + a_2{\alpha} + \cdots + a_{n-1}{\alpha}^{n-2} + {\alpha}^{n-1})$$
we can see that ${\alpha}\beta = 1$, so ${\alpha}$ has an inverse in terms of ${\alpha}, {\alpha}^2, {\alpha}^3, \ldots$ Also, ${\alpha}^n = -(a_{n-1}{\alpha}^{n-1} + \cdots + a_1{\alpha} + a_0)$, so we get that $F({\alpha})$ is just the $F$ vector space with basis $\{1,{\alpha},{\alpha}^2,\ldots,{\alpha}^{n-1}\}$.
In the case that ${\alpha}$ is transcendental, we get that $F(\alpha) \cong F(t)$, the field of rational functions over $F$