Here's a paragraph from the book:
How is he going from $z\neq 0$ to allowing $0$ in the last entry (e.g. $[1:m:0]$)? One way I can see is by taking the Zariski closure of the set since the set was a plane minus a line through the origin union the origin. But why are we doing it?
Edit: I shouldn't use the term 'algebraic closure' in this context
A point $(x,y)$ in $\Bbb A^2$ corresponds to a point $[x:y:1]$ in $\Bbb P^2$. In the projective completion of the affine line, the point at infinity has the last coordinate equal to $0$.
This is also valid for any algebraic curve. For instance the unit circle $x^2+y^2=1$ in the affine complex plane has two points at infinity $[1:\pm i:0]$ in its projective completion, since its homogeneised equation is $x^2+y^2=z^2$.