The big reason is to provide an (automated) vertical path / guidance that the FMS can follow, even with older DME/DME based equipment.
FMS Bridge visual is basically an overlay of Quiet Bridge Visual. It was supposed to provide reduced workload to the controller, as the waypoints and stepdowns altitudes are coded into the FMS.
FMS Bridge visual is considered a special procedure, (and it is an instrument approach) which the airline and pilots are required to be approved to use. This has been discussed at length in this thread.