Nice to see you found my link. To pass more current to your LEDs, you can use individual transistors or a chip which provides six or eight transistors in a single package.
The way this approach works is that the transistor is connected in the current path of each led on the separate power circuit. an Arduino output pin is connected to the base of the transistor. When the arduino pin is low, the transistor will act like a switch in the off position. If the Arduino pin is high the transistor will act as if it is a switch in the on position. This switching between on and off works fast enough for the PWM/analogWrite that you are currently using for brightness and colour mixing.
If you search for 'transistor as a switch' you will find approaches using individual transistors, if you search for 'darlington driver' 'hex driver' or 'transistor array' you will find approaches using chips.