Fox had introduced the original 35mm version of CinemaScope in 1953 and it had proved to be commercially successful. However the additional image enlargement needed to fill the new wider screens that had been installed in theatres for CinemaScope resulted in visible film grain. The obvious solution was to use a larger film so that less enlargement would be needed. CinemaScope 55 was the result of this thinking, and was one of three "High Definition" film systems introduced in the mid '50s, the other two being Paramount's VistaVision and the Todd-AO 70 mm film system.
Fox determined that a system that produced a frame area approximately 4 times that of the 35mm CinemaScope frame would be the optimum trade-off between performance and cost, and the 55.625mm film width was chosen because it provided that. Camera negative film had a larger grain than the film stocks used for prints so there was some logic in using a larger frame on the negative than on prints. Since prints need to allow space for soundtracks whilst camera negative doesn't CinemaScope 55 had different frame dimensions for camera negative and print film.