Intermediate fields of a finite field extension that is not separable

To see why there are infinitely many intermediate fields between $K=\Bbb{F}_p(x^p,y^p)$ and $L=\Bbb{F}_p(x,y)$ you can do the following.

Let $z$ be any element of $K$. Consider $w=x+zy$. We see that $w^p=x^p+z^py^p\in K$, so $K(w)$ is a degree $p$ extension of $K$. If $w'=x+z'y$ for some $z'\in K$, $z'\neq z$, then we easily see that $K(w,w')=L$. Therefore different choices of $z$ lead to different intermediate fields. The element $z$ can be chosen in infinitely many ways, and the claim follows.

IMHO the standard way of showing that $L/K$ is not simple is to observe that any element $u\in L$ has the property $u^p\in K$. Therefore $[K(u):K]\le p<[L:K]$.