Finally a clear picture that shows the problem.
Basically, your wiring to the RJ45 breakout board is wrong.
Connect the motor wires as follows:
Black to Pin1
Green to Pin3
Blue to Pin5
Red to Pin7
If the motors move, then that is your problem.
Then, you need to put a small jumper wire between the following pins
on the screw terminals:
1 to 2
3 to 4
5 to 6
7 to 8
That way, the signal is split between each two adjacent wires on the cable.