The conformal group of $S^n$.

Let's say you want to find all locally conformal maps on some open subset of $\mathbb{R}^n$ where $n\geq 3$. The case of $n = 2$ is rather special, any holomorphic function with nonzero derivative is locally conformal.

Sticking to the case $n\geq 3$, unwinding the definitions leads to a system of PDEs which can be explicitly solved. This is known as Liouville theorem. One class of solutions cannot be extended to the whole $\mathbb{R}^n$ - these are the spherical inversions. Thus one is led to consider the conformal compactification of $\mathbb{R}^n$ - the sphere $S^n$, where the spherical inversions are defined on the whole space. Conformal compactification means that we can embed $\mathbb{R}^n$ into compact $S^n$ and that the embedding is conformal map (in this case it is the inverse of the stereographical projection). Now we know from the Liouville theorem that any locally conformal diffeomorphism of the sphere is either translation, rotation, dilatation or spherical inversion. The maps are quite explicit on $\mathbb{R}^n$. To get the equations on the sphere you have to "conjugate" it with the stereographical projection which is also quite explicit.

In fact, one can describe explicitly isomorphism between the group of conformal diffeomorphisms of $S^n$ and the linear Lie group $\mathrm{SO}(n+1,1)$. For proof of the Liouville theorem and for details on this isomorphism see notes by Slovák, page 46 onwards.

Try the book A Mathematical Introduction to Conformal Field Theory by Martin Schottenloher. Chapters 1 and 2 go over some of the proofs you are looking for and the book is example driven.

Try Lecture One of Eastwood, "Notes on Conformal Differential Geometry" (http://dml.cz/dmlcz/701576).