DRV8825 Stepper Driver Burning Up
First of all let me note something: I do not trust theese little ICs for driving my steppers. If you look at the datasheet of the  NEMA 23 steppers, you will see some graphs indicating the current needed to operate the coils inside the steppers:

is 1.4 Amps per phase is too much for your DRV8825 stepper driver, even though its datasheet says it can carry 2.5A, you need to get a bigger/stronger stepper driver. I had a issue where my stepper drivers got burned twice because they were not strong enough for the currents they needed to handle.
So I think its really likely your ICs (or one of them) burned already, creating a short and drawing current constantly, overheating the wire untill it melts.
I recommend connecting only one IC, with one stepper, see if it works, then move to soldering the second one.
Use thicker wires if possible.
Do not forget to clean your board after soldering, using isopropanol. Flux leftover on your pcb will conduct, leading in shorts.
When soldering, try to use lower temperature so that you wont fry your IC while soldering.
Also, I would recommend using female header pins on the pcb in place of the driver ICs

And then place your ICs on the headers. That way, you wont have to solder directly on the IC, and you will be able to remove/replace the IC.