Because the Riemann mapping theorem does not hold in higher dimensions. While there are all sorts of conformal mappings in dimension 2, for higher dimensions Liouville's theorem restricts all possible conformal mappings to the ones that are compositions of similarities, translations, orthogonal transformations and inversions. In generality there are contractible spaces not homeomorphic (therefore not conformal) to $\mathbb{R}^n$ such as Whitehead continuum
As for the proof of Liouville's theorem, maybe this article is of interest where you can see a sketch of Nevanlinna's original proof and a proof by nonstandard analysis.