The easiest way to control OpenSegment is over serial. You need only 3 pins: PWR, GND and RX. The RX pin on OpenSegment should be connected to the TX pin of your microcontroller.
The easiest way to physically connect to OpenSegment is through a 3-pin JST connector.
Solder the JST onto the backpack, plug the JST cable in, and then plug the red wire to 5V to 7V, black wire to GND, and yellow wire to the TX pin of your development board.