I'm studying complex analysis and Möbius transformations are introduced.
I understood that those class of mappings are recurring when studying conformal mappings of $\mathbb{C}$ (conformal mappings of the unit disk are Möbius transforms, same for $\mathbb{C}$ extended with the point at infinity, etc.).
I also read that those transformations may be defined using the action of PGL(2, $\mathbb{C}$) on $\mathbb{C}$, a definition that makes their group structure much understandable.
When reading Complex analysis books, it seems like it just happens that PGL(2,$\mathbb{C}$) induces conformal mappings on the Riemann Sphere (because those induced functions happen to be holomorphic). I guess this doesn't just happen randomly and it should be part of a larger theory which explains why elements of PGL(2,$\mathbb{C}$) define conformal maps.
If so, what is this theory and how could that connection be summarized ?
We can examine the conformal automorphism group of $S^n$ and then examine $n=2$ as a special case, where $\mathrm{PGL}_2\mathbb{C}$ is relevant by way of an "accidental isomorphism."
Consider the pseudo-Euclidean space $\mathbb{R}^{p,q}$. Its projectivized null cone is $(S^{p-1}\times S^{q-1})/\mathbb{Z}_2$. Thus, we can identify $S^n$ with the projectivized null cone of $\mathbb{R}^{1,1+n}$. This is acted upon by the split orthogonal group $\mathrm{O}(1,1+n)$. IIRC, the Cartan-Dieudonné guarantees this is generated by pseudo-Euclidean reflections, which correspond to inversions in $S^n$ (a la inversive geometry, where $\mathbb{R}^n$ and $S^n$ can be treated interchangeably via stereographic projection). That these tranformations are conformal then follows from the inversions being conformal, which is arguably easier to see in affine space (and also stereographic projection is also conformal). The theory behind all this is called "Möbius geometry."
[This generalizes to "Lie sphere geometry": the set of hyperspheres in $S^n$ can be identified with the projectivized null cone of $\mathbb{R}^{2,1+n}$, on which $\mathrm{O}(2,1+n)$ acts. But these transformations can turn points on $S^n$ into hyperspheres of nonzero radius.]
For $p+q\le 6$ there are a patchwork of "accidental isomorphisms" between spin groups $\mathrm{Spin}(p,q)$ (double covers of $\mathrm{SO}(p,q)$s) and classical matrix groups over composition algebras - reals, complex numbers, quaternions. There are even more of these if we are willing to be more flexible, e.g. use split complex numbers, split quaternions, octonions and split octonions. The most important one is arguably the one $\mathrm{SL}_2\mathbb{C}=\mathrm{Spin}(1,3)$, relevant to relativity (check out the signature!), which restricts to an equally important one $\mathrm{SU}_2\mathbb{C}=\mathrm{Spin}(3)$, relevant to quantum theory.
It remains to understand how $\mathrm{SL}_2\mathbb{C}$ is relevant. We know $\mathrm{PGL}_2\mathbb{C}$ acts on the complex projective line $\mathbb{CP}^1$ as automorphisms. An element of $\mathbb{CP}^1$ may be represented by a vector $v\in\mathbb{C}^2$, which corresponds to a hermitian matrix $vv^\dagger\in\mathfrak{h}_2\mathbb{C}$ (the corresponding orthogonal projection). This matrix is well-defined up to scalar multiplication, and has determinant zero, so this actually identifies $\mathbb{CP}^1\simeq S^2$ with the null cone of $\mathfrak{h}_2\mathbb{C}$ (which uses the determinant as a real pseudo-Euclidean "inner product," of signature $(1,3)$). We can restrict the action to $\mathrm{SL}_2\mathbb{C}$, since scalars act trivially. Also relevant is that $\mathbb{CP}^1$ may be identified with the Riemann sphere $\mathbb{C}\sqcup\{\infty\}$; if we transport the action of $\mathrm{SL}_2\mathbb{C}$ we get the usual action by Möbius transformations.