Book on quasiconformal mappings?

Use Lectures on Quasiconformal Mappings by Ahlfors. The second edition is TeX-ed and has some survey articles attached to it which are more advanced than the main text, but give an insight into how quasiconformal maps are used in complex dynamics and hyperbolic geometry. The motivation and history is a part of what you'll learn reading that book.

A standard reference for quasiconformal mappings in $\mathbb{C}$ is Lehto/Virtanen, "Quasiconformal Mappings in the Plane" - containing also quite some historical references. For the higher-dimensional theory of quasiconformal mappings, I'd refer to Väisälä's book "Lectures on $n$-Dimensinonal Quasiconformal Mappings". More a modern approach via PDEs, the book "Elliptic PDEs and Quasiconformal Mappings in the Plane" of Astala et. al. is the right source for your self-study, containing a lot of historical remarks as well.