Function Field of projective space

The function field of $V$ is defined as the field of fractions of $K[X]/I(V)$ for affine varieties $V$. In the case of projective varieties, Silverman chooses a Zariski-dense affine open subset $V$ of the variety and defines the function field of the variety as the function field of the subset $V$. Of course, one can prove it is independent of the choice of $V$. When the variety is $mathbb{P}^n$, choose $V = mathbb{A}^n$ and so $I(V)={ 0 }$, so $K(mathbb{P}^n) = K(V) = K[X]/{ 0 } = K[X]$, where $X$ is shorthand for $X_1,ldots,X_n$. Finally to get an isomorphism of the subfield of $K(X_0,ldots,X_n)$ you described with the field I just described, just set $X_0=1$.