To start off, you will need to decide how you want to power the A-Star 32U4. If you want the entire robot (e.g. the motor driver and power distribution board and A-Star 32U4 Prime) to both be controlled with a single power switch, you can connect power from any of the VSW pins on the motor driver and power distribution board to the VIN pin on the A-Star 32U4, as well as a ground connection between the two boards. Please note that this will bypasses the reverse voltage protection on the A-Star 32U4 Prime.
As far as controlling the motors, the PWM, DIR, and SLP pins (for each motor) on the motor driver and power distribution board are available in the space between where the motors line up. If you do not need the SLP pins, you could use our DRV8835 Motor Shield Arduino library to control the motors. The additional connections you would need to make would be DIR and PWM for the left motor to pins 7 and 9 on the A-Star 32U4 Prime respectively, and DIR and PWM for the right motor to pins 8 and 10 on the A-Star 32U4 Prime.
If you are using the Romi Encoders, the ML+, ML-, MR+, and MR- terminal block connections can be left disconnected; these are intended to be used for connecting the motor to the motor controller, but the Romi encoders handle this connection when plugged directly into the motor driver and power distribution board, as shown in this picture:
To access the encoder signals, you can make connections from the ELA, ELB, ERA, and ERB pins to the relevant pins on your A-Star 32U4 Prime (which pins are relevant will depend on your code). Please note that since the motors and encoders sit above these pins, there might not be room for headers, so you might consider soldering your wires directly to the pins. Alternatively, you might be able to solder a header in at an angle.
If you are unsure about any of the connections, you can post a proposed wiring diagram here, and I would be happy to take a look.