Let $L|K$ be an extension of number fields and consider the corresponding (integral) extension of ring of integers: $R_L|R_K$. Note that $R_L$ and $R_K$ are finitely generated over $\mathbb{Z}$, hence they are free $\mathbb{Z}$-modules. Then if $Q\subset R_L$ is a nonzero prime ideal, and we set $P:=Q\cap R_K$, then $R_L|Q$ is a finite field (being a quotient of $(\mathbb{Z}/p\mathbb{Z})^n$ if $R_L\cong\mathbb{Z}^n$ and $p\mathbb{Z}:=Q\cap\mathbb{Z}$). In particular, $(R_L|Q)|(R_K|P)$ is separable.
I can't understand the bold part, in particular when he says "a quotient of"
My best guess is that the author means the following.
You have probably proved earlier in the course that $R_L\cong\mathbb{Z}^n$ as an abelian group. Because $p\in Q$, and $Q$ is an ideal, it then follows that $$ pR_L\subseteq Q\subset R_L. $$ This implies that $$ R_L/Q\cong (R_L/pR_L)/(Q/pR_L). $$ But here $R/pR_L=(\mathbb{Z}^n)/p(\mathbb{Z}^n)=(\mathbb{Z}/p\mathbb{Z})^n,$ so we have just written the field $R_L/Q$ (it's a domain because $Q$ is prime, and as pointed out by Billy, it's then a field) as a quotient of $(\mathbb{Z}/p\mathbb{Z})^n$.