An elementary proof is the following: first you prove that every holomorphic map $f:S^2\to S^2$ is a rational function. Indeed, such map must be a meromorphic function, it has finitely many zeros and poles and is either regular or has a pole at $\infty$. So the degree of $f$ is defined: the equation $f(z)=a$ has $d$ solutions, counting multiplicity, for every $a\in S^2$. Therefore, if the map is bijective, we must have $d=1$ that is $f\in PSL(2,C)$.

This proof is based on elementary algebra, no analysis is used.

An elementary proof is the following: first you prove that every holomorphic map $f:S^2\to S^2$ is a rational function. Indeed, such map must be a meromorphic function, it has finitely many zeros and poles and is either regular or has a pole at $\infty$. So the degree of $f$ is defined: the equation $f(z)=a$ has $d$ solutions, counting multiplicity, for every $a\in S^2$. Therefore, if the map is bijective, we must have $d=1$ that is $f\in PSL(2,C)$.

This proof is essentially elementary algebra, only one fact from Analysis is used: the "Fundamental theorem of algebra".