Connect an Arduino directly to RGB LED strip
Those LED strips are multiple parallel chains of 3 LEDs in series, with a 150Ω resistor for each chain.  The chains are arranged as common anode, and the chains are grouped into the three colours for the cathodes.

You require a power supply with enough current to supply all the LEDs (such as the 3A one supplied in the package).

Further, you require a method of switching each of the three channels, such as n-channel MOSFETs, which can handle enough current for each channel (one MOSFET per channel) - that is, 1A per channel if the whole strip takes 3A (but you can "derate" it by providing capacity for more current which helps alleviate some heat dissipation problems).

As long as the MOSFETs are logic level (such as the IRLxxx series) you can then switch them directly using the PWM outputs of the Arduino.