What's the point of buying the remote transmitter and controller box if you are planning on controlling the strings from the arduino?
From the picture of the controller box, it looks like the connector may have 3x 12V,2A out and 1 common ground.
So if you're going to put your own circuit in place of that, you need 3 decent P-channel MOSFETs to provide the 12V connection under PWM control for brightness control, fading, color mixing etc.