For a more detailed explanation, with tons of great pictures, go here: http://bulatov.org/math/1001/
Here's the short version: there's a way to measure distances on a disk that makes it into a model of the hyperbolic plane. There are actually a number of ways, but Bulatov - and Escher - use the Poincare disk model, because in this model straight lines look like portions of circles: very pretty. Then, according to the Riemann mapping theorem you can map this disk in a conformal way onto a disk with 4 slits cut out. The hard part is finding a formula for how to do it, and then implementing it on a computer.
For more details, try these picture-packed pages:
and this more advanced one: