It uses very stable dyes. The original transparencie is copied three times through three "color seperation filters" resulting in three negatives the size of the finished print. These in turn are contact printed onto a special film which has a rather thick emulsion designed to soak up dye. Every step has to be done with a pin registration board to insure that the three colors print in register with one another. Each of the three positives
is soaked with dye and which is then transferred to the paper.
The process gives you a lot of room to play with the image or really screw up since you can seperately burn and dodge each of the three negatives or change their contrast a bit in the developer.
Years ago they used to make big unwieldly cameras which would simultaniously expose three sheets of film in standard sheet film holders. The interior of the camera had several partially silvered mirrors along with the three filters to send the light to the three sheets of film.
Early Technicolor motion picture cameras worked the same way, with three magazines each with its own roll of film, and the final print is made by transferring dyes from the negatives. Since the color in the print is so stable it's still a favored way of making release prints, but the seperations are made from the original color negative.
Kodachrome is made by adding each dye seperately during the processing while other color films have the dye already in the emulsion, and it couples with the silver image during processing. At one time Kodak would make you Kodachrome prints from your slides using the same stable dyes.