Dimension of a point in a polynomial ring.

Question

Let $k$ be an algebraically closed field. Let $X \subset \Bbb A^n$ be a nonempty variety.
Now suppose $X$ is a point then it easy to see that $\dim_k k[X] $ is finite. Here $$k[X]:=\{f \in k[X_1,X_2,\dots,X_n \;| \; f \text{ is a polynomial function} \}$$ This is from Algebraic Geometry (Klaus Hulek), I'm confused why is this easy to see. Any explanation would much much Appreciated. Thanks!