$((S_{\bullet})_f)_0$ means the $0$-graded piece of the $\mathbb{Z}$-graded ring $(S_{\bullet})_f$
In Vakil's notes page 149, it seems that the author considers a construction $Spec((S_{\bullet})_f)_0$ as affine building blocks of Proj construction, and he remarks that
As motivation for considering this construction : applying this to $S_{\bullet}=k[x_0,...,x_n]$, with $f=x_i$, we obtain the ring appearing in (4.4.9.1):
$k[x_{0/i},x_{1/i},...,x_{n/i}]/(x_{i/i}-1)$
I couldn't see why we could obtain the ring $k[x_{0/i},x_{1/i},...,x_{n/i}]/(x_{i/i}-1)$ if we take the special case as him. Could you give me some hint? Thanks in advance.
An element of $((S_\bullet)_{x_i})_0$ is of the form $f=\frac{P(x_0,...,x_n)}{x_i^d}$ where $P$ is a homogeneous polynomial of degree $d$. So if you define $y_j=\frac{x_j}{x_i}$ you can write $f=Q(y_0,...,y_n)$ where $Q$ is a polynomial. For example you can write $$\frac{x_1x_2^2+x_1x_2x_3}{x_i^3}=y_1y_2^2+y_1y_2y_3.$$ Of course $\frac{x_i}{x_i}=1$ so $Q$ is realy an element of $$\frac{R[y_0,...,y_n]}{y_i-1}.$$ Conversely if you have an element of this ring you can multiply monomials by $y_i$ untill you get a homogeneous polynomial and then change your coordinates back to $x_i$. For example $$y_1^2+2y_1+3=y_1^2+2y_1y_i+3y_i^2=\frac{x_1^2+2x_1x_i+3y_i^2}{y_i}.$$ If you see the usual construction of $\Bbb P^n$, these are the usual change of coordinates and rings so this is just a way to do that construction which can used in more general settings (for example you can see definition of blow-up).