We are not familiar with that device and cannot tell you exactly how all of the pins work. It is possible that the motor is a brushed DC motor. If so, the outputs of our brushed DC motor drivers or controllers could probably be connected to the +Motor and -Motor pins to make the motor move. The rest of the pins appear to be for sensors (like encoders and limit switches) integrated into the device to determine the position of the stage. Generally, sensors like that need a power supply, which could be what the +5V and Ground pins are for.
We do not have an out of the box solution that would work with all of logic signals that connector appears to provide. Our RoboClaw motor controllers can use feedback from a quadrature encoder to control the position of a brushed DC motor, but you might still need to use something like a microcontroller to monitor the state of any limit switch outputs and control the RoboClaw appropriately. Alternatively, you could use a microcontroller with your own custom firmware to monitor all of the appropriate sensor pins and one of our brushed dc motor drivers to provide variable power to the motor. Our TB67H420FTG Dual/Single Motor Driver Carrier board should work well with the voltage and current indicated in the picture you posted.