It's magic basically, but if you want the scientific answer...

The process is called Dye Sublimation. Artwork is printed onto a sheet of high-release paper and transferred onto apparel using heat and pressure. Heat converts the solid dye particles into a gas — known as sublimation — and bonds them to the polyester fibers. Unlike screen printing, the dye is absorbed by the fabric for a fantastic soft-to-the-hand feel. 

Care Instructions for Dye-Sublimated Apparel:

Machine wash inside-out with cold water and tumble dry on low heat or line dry. For best results use mild detergent and avoid the use of bleach, fabric softener, and dryer sheets.