Click on the applet and press z to begin painting. You may want to press F11 to view in fullscreen mode. If you want to save your picture, press the "Print Screen" button. Then open up Paint (Start menu->programs->accesories->Paint) and press ctrl+v to copy. I recommend testing this before making any fancy pictures.
Don't forget the much cooler Rainbow Paint 2
Would you like to see the source code or download the program? You can get the files you need here.
UW Madison >> CS department >> Tim's website >> RainbowPaint