According to the link at your wiki the RGB LEDs are rated for 2 to 3.2 volts. Your instructions for connecting the LED indicate connecting the LED to 5 volts on the Arduino Nano. I'm not sure why the LEDs don't burn out rather quickly when powered with 5 volts. Maybe you are doing something in the sketch to limit the current. The LEDs apparently come with resistors which could be used to limit the current so the LEDs do not burn out, but these resistors are apparently not included in your kits.
The Arduino Nano has a 3.3 volt pin which could be used instead of the 5 volt pin. Using that pin might protect the LEDs, but using one of the resistors on the power leg of the LED might also be a good idea.