I was reading Ravi Vakil's notes on his website and he states the Hilbert Nullstellensatz (3.2.5.): If $k$ is any field, every maximal ideal of $k[x_1, ..., x_n]$ has residue field a finite extension of $k$. Translation: any field extension of $k$ that is finitely generated as a ring is necessarily also finitely generated as a module (i.e., is a finite field extension).
I understand (at least I think I do) the statement of the theorem, but I just don't understand why this statement of the theorem translates to what he wrote in "Translation". Could someone please explain me how this works? Thanks!