What you've bought is just an Arduino with a set of RGB LEDs. There is not much robot specific about this kit. I'd start by completing some simple testing without connecting it to your robot.
There is some decent documentation on the AndyMark GitHub page but I think the tutorial on Adafruit is better to get you going:
https://learn.adafruit.com/digital-led-strip
The wiring for these is pretty common. Basically you just need to get 5v power to the Arduino and LED strip. Note that the LED strip can not be powered from the Arduino it self.
Once you have it wired up there is sample code for this kit directly from AndyMark:
http://files.andymark.com/PDFs/am-3010+ExampleCode.pdf
Once you have it working off your robot and your comfortable with the code you could go ahead and just wire it into your robots 5v supply and leave it at that. If you want to have the LEDs react to something on your robot search the CD forums. It's very doable but it can get complicated.
Here is a simple setup I happen to be working on. The unplugged bit is a 5v 2amp wall plug.
