You could use the Lasso Select tool to select the item that you want to cut out and put on another image, and the checkerboard means that the background is transparent. I think you should open your picture, and add a new layer, and add the item that you cut out to the new layer, then change the layer's properties to "Overlay". I think it should come up. If not, just check out this tutorial, it shows how to cut out images:
http://paintdotnet.12.forumer.com/viewtopic.php?t=3434
Hope it helps