How to change breakout board

[Dragonfly]

NC Studio card sends signals via a 15 pin DB15 connector. Taking out the card from the PC is not compulsory but I'd suggest removing it. You have to unplug the three orange connectors and follow their wires to the drivers to determine the correspondence to Step and Dir terminals.
As far as I can see on the photo yellow is +5V, green is Dir(ection), Blue is Step Pulses. All three sets seem to be combined on JP4 connector and the one next to it.
The rest have to be traced to where they lead. Some of them are definitely for input signals (like limit switches) back to the PC.
Otherwise connections a clearly described in the manual. After identifying which wire is for what there shouldn't be problems to connect them to the new BoB. Without connecting any inputs, only the motors control wires and correct setting outputs in "ports and pins" menu you should be able to do a test move of the machine.
Currently your drivers are set to trigger on pulse going down which should be set accordingly in Mach 3. "Active low".
Disable E-stop input in Mach3 and you can make a test.

Edited because didn't identify the motor wires correctly on the photos at first look.