Why do the maximal the ideals of $\mathbb C[x_1,\dots,x_n]$ that contain $I$ correspond to the maximal ideals of $\mathbb C[x_1,\dots, x_n]/I$?
In Artin, theorem 11.9.1, Artin says this result is by the correspondence theorem. How does this follow?
The correspondence theorem says the ideals of a ring $R$ correspond to the ideals of $R/I$. It doesn't say anything about maximal ideals.
Let $M$ be maximal in $R$ and assume the ideal $M/I$ in $R/I$ is contained in another ideal, then by correspondence theorem, $M/I\subseteq S/I$ for some ideal $S$ in $R$ such that $M\subseteq S$. The maximality of $M$ in $R$ shows that $S=R$, so $S/I=R/I$, this proves the maximality of $M/I$ in $R/I$.