Any element of $PGL(2,\Bbb{C})$ determines a map $$\Bbb{P}^1\to\Bbb{P}^1$$ $$[z:w]\to [az+bw:cz+dw]$$ in other words, a Mobius map $$[z:1]\to [\frac{az+b}{cz+d}:1]$$
How are the two maps equivalent? How can we just assume that $w=1$ and that $cz+d\neq 0$?
$\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$In the projective line, an arbitrary quotient $Z/W$ of complex numbers (not both zero) makes sense: If $W \neq 0$, then $z = Z/W$ has its ordinary meaning as a complex number. If $W = 0 \neq Z$, we write $Z/W = \infty$. It's straightforward to check that some arithmetic operations with $\infty$ can be performed consistently with the familiar rules of complex arithmetic; particularly, $Z/\infty = 0$ for all $Z \neq \infty$. This means we can meaningfully form quotients $Z/W$ so long as $(Z, W) \neq (0, 0)$, even if $W = 0$.
As a set, the projective line may be viewed as $\Cpx \cup \{\infty\}$. Alternatively, the projective line may be viewed as two copies of $\Cpx$ with respective coordinates $z = Z/W$ and $w = W/Z = 1/z$. The origin in each copy of $\Cpx$ is the point at infinity in the other copy, see Flag manifold to Complex Projective line.
Multiplication by a non-singular $2 \times 2$ complex matrix defines a bijection of the space of non-zero pairs of complex numbers, and induces a bijection of the space of lines through the origin of $\Cpx^{2}$. Particularly, if $ad - bc \neq 0$, then $$ \frac{aZ + bW}{cZ + dW} = \frac{az + b}{cz + d} $$ makes sense for all $(Z, W) \neq (0, 0)$, i.e., the numerator and denominator are not both $0$. If the denominator is $0$, the quotient is $\infty$.
Every (complex) line through the origin contains some point $(Z, W) \neq (0, 0)$. If $W \neq 0$, then $$ (Z, W) \sim (Z/W, 1) = (z, 1). $$ If $W = 0$, then $Z \neq 0$, so $(Z, 0) \sim (1, 0)$, which may be viewed formally as $(\infty, 1)$. Writing everything in homogeneous coordinates, we have \begin{gather*} [1 : W/Z] = [Z: W] = [Z/W: 1], \\ \left[1: \frac{cZ + dW}{aZ + bW}\right] = \left[aZ + bW: cZ + dW\right] = \left[\frac{aZ + bW}{cZ + dW}: 1\right], \end{gather*} with the understanding that any quotient with denominator $0$ represents the value $\infty$.
