24v LED strip with common anode?
Your strip has 2 LEDs with different color temperature directly besides at the same spot for every element of the strip. So the adjusting of the overall color temperature is done by changing the relative brightness of both LEDs. If the LED with the warmer light is brighter, the overall color temperature will also be warmer. Common Anode means, that V+ is the connected to the positive pin (anode) of both LEDs, so it has to be connected to +24V. The electrical circuit is completed by connecting the pins W and C (Warm and Cold) to ground. To control the individual brightness of the warm and cold LED groups, you have to use PWM.

You are right, that an Arduino cannot handle currents this big. You need a transistor for this. I would suggest using a MOSFET, since this is the easiest and most efficient possibility. By one MOSFET for every channel (Warm and Cold), which is capable of handling the needed power (Important! Or you will fry the MOSFET). Also it should be a logic level MOSFET, that is well in saturation of the gate voltage for the output voltage of your Arduino. Connect the MOSFETs Drain pin to the channel on the strip, the Source pin to ground of your power supply and the Gate pin to a PWM pin of the Arduino. Also connect the ground of the power supply with the ground of the Arduino.

Now you can control the brightness of the Warm and of the Cold LEDs by adjusting the PWM duty cycle with the Arduino.