Looking at your pictures I would bet that your ESCs are actually a fairly standard configuration. Three control wires. There just isn't any real justification for Phantom or any of the other drone makers to not buy a standard design. In your case one of the three standard wires is doubled for some reason. The connector back to the main board is a four pin connector with only 3 pins populated. My guess would be that the darker brown wire is ground, the adjacent wire is the PWM signal and the next one is + 5V. You could do a little verification of this if you have retained the original battery leads for the Phantom. Just measure continuity between the dark brown wire (when everything is plugged together) and the battery negative.
With this information and reading the Arduino documentation for the SERVO you should be able to get electrically hooked up. Note that the motors might turn if powered just by the Arduino 5V, but they really need their own high current supply. This is supplied by the wire on the corner opposite the signal connection. Since they didn't use a standard color code you will have to find continuity to the ground to see which one is positive and which is negative.
As others have said before, most ESCs have a safety feature. They won't turn on until a sequence of throttle commands is sent. This helps prevent an unexpected spinning propeller when the battery is plugged in. This feature has reduced but not eliminated the parade of gory pictures of people with hands, arms, stomachs or faces chewed up by a poor quality circular saw.
A typical sequence is to require a zero throttle position, followed by full throttle, followed by a return to zero throttle before arming and making the motor operate. Usually once armed a few cycles of a sine wave is sent to the motor to make it beep warning you that it is armed and dangerous.
Different makers use different interpretations of full throttle. Some use max pulse width, others use minimum. This combined with variations in the arming sequence mean that trial and error to find the right sequence will take a while. Again, google for ESCs and read the manuals for a few to get some ideas for sequences to try. You might try googling phrases like "Phantom motor arming" and the like to find specific directions, although on drones this is likely done in the control software and considered proprietary.