Help with this proof in Fulton book

Help with this proof in Fulton book

I'm really confused with the definitions of coordinate rings and field of
rational functions. I'm trying to understand this proof which I was stuck
in the very beginning:

First I didn't understand the definition of $J_f$. we have $\overline
G=G+I(V)$ and $f=f_1+I(V)$, where $f_1$ is the residue of $f$ in
$\Gamma(V)$. The $\overline
Gf=\bigg(g+I(V)\bigg)\bigg(f_1+I(V)\bigg)=\bigg(gf_1+I(V)\bigg)$, so this
multiplication is not always in $\Gamma(V)$, since $\Gamma(V)$ is by
definition $k[x_1,...,x_n]/I(V)$?
Second I didn't understand why the points of $V(I_f)$ are exactly those
points where $f$ is not defined.
I really need help to understand this proof.
Thanks in advance.