Let $X \subset P^n$ be some projective variety, and $I$ be its ideal. We consider the 0th graded piece of the localization $(I \cdot k[Z_0,...,Z_n,Z_i^{-1}])_0$ in the coordinate ring $(k[Z_0,...,Z_n,Z_i^{-1}])_0 \cong k[z_1,...,z_n]$. This is exactly the ideal of the affine open subset $X\cap A^n \subset A^n$.
I cannot understand why the 0th graded piece of the localization $(I \cdot k[Z_0,...,Z_n,Z_i^{-1}])_0$ in the coordinate ring $(k[Z_0,...,Z_n,Z_i^{-1}])_0 \cong k[z_1,...,z_n]$ is exactly the ideal of the affine open subset $X\cap A^n \subset A^n$. (specifically, I do not know why this is called localization)
The isomorphism $(k[Z_0,...,Z_n,Z_i^{-1}])_0 \cong k[z_1,...,z_n]$ is just the dehomogenization with respect to $Z_i$. Each $F\in I$ of degree $r$, corresponds to $F/Z_i^r \in (I \cdot k[Z_0,...,Z_n,Z_i^{-1}])_0$, hence corresponds to its dehomogenization with respect to $Z_i$.
$k[Z_0,...,Z_n,Z_i^{-1}]$ is the localization of the ring k[Z_0,...,Z_n] with respect to the multiplicative set $\{1, Z_i, Z_i^2, \dots \}$. This is a basic construction of Commutative Algebra.