What is the grey and white checker board in paint.net

This grey and white "checker board" is simply a blank space. The initial "white" background on the canvas is not blank space.

What is the point of blank space you ask?

Look at the icons on your desktop. You will notice that there is a graphic, but then the desktop background envelopes the graphic. This is because of blank space.

Without that blank space it would look like this:

Notice the white space instead of the blank space. Saving images as JPGs or GIFs do not recognize blank space. Saving images as PNGs do recognize blank space.

You could save an image of a blue circle surrounded by blank space as a PNG and when you upload that image to the web, all you will see is the blue circle with the background of the website behind it - no white corners.