How can I get GIMP to de-select the background in my attached picture? Everytime I try to use Select None...nothing happens.
Basing upon your image, it seems that you are referring to the layer boundary, which is the yellow dotted line that surrounds a layer in the image window (the selection boundary is grayed and keeps moving, like "marching ants"):
You can hide the layer boundary by unchecking the flag Show Layer Boundary in View menu:
It should do the trick:
I don't think you've actually got a selection in force there!
The dots around the oar-like shapes are not GIMP's "marching ants" (which would indicate a selection) but something else - individual pixels and pairs of adjacent pixels either placed onto the image or left-over from some previous step. And they are not anti-aliasing pixels.
I think the most likely explanation is that those dots are artefacts which have been left by a less-than-perfect extraction of the shapes from some previous background.
However you got to this point, if the dots aren't on a separate layer, to extract the shapes onto a transparent background (which I presume is what you are trying to do) you will first need to erase the dots to white and then use Colours > Colour to alpha (white) to change the background to transparent.