Koebe–Andreev–Thurston theorem (known also as the circle packing theorem) says that any planar graph can be realized by a set of (interior-) disjoint disks corresponding to vertices, such that two discs are tangent iff the corresponding vertices are connected to each other.

Where can I find the/a proof of this theorem, and what should I learn to understand it?

I prefer proofs which are elementary, but other proofs are welcome too.