The Eye Dropper (or Dropper) isn't a painting tool. It enables you to pick up a color from any image and set the foreground/stroke or background/fill material color for use with other tools like the Airbrush, Paint Brush, and Flood Fill tool.
Sample sizes can be 1x1, 3x3, 5x5, 7x7, or 11x11 pixels, as seen in the Dropper tool options here. The Dropper is useful when you need to precisely match a specific color.
Open the colorshapes image in PSP, then select the Eyedropper tool. Left click the Red Square. This will set your foreground/stroke color to the same shade of red: RGB 255/0/0. With the Eyedropper tool still selected, right click the Yellow Triangle. This will set your background/fill color to the same shade of yellow.
TIP: You can temporarily switch to the Eyedropper tool from many other tools by holding down the control key.