I've been looking through existing forum questions regarding the problem of breaking a bitmap down into smaller parts, but don't quite get how I should do it. I'm currently trying to separate the objects in the red rectangles from the rest of the bitmap.

I'm on Windows using the latest version of Inkscape, just using a basic mouse and keyboard
 

I should mention that Break Apart (Ctrl + Shift + K) does not give the desired result, instead turning the bitmap all black for some reason:

I'd turn off the fill and see how it looks.

How do I turn off the fill?

Use the Fill & Stroke panel to add or remove fill colours, stroke colours and change opacity.

There's some information here...

https://inkscape.org/doc/tutorials/basic/tutorial-basic.html

Scroll down to the Fill and Stroke section.

I tried turning off the fill, and setting it to "None", but it still doesn't solve the issue.

The fill is still set to have color. I'd click the X in the fill tab of the dialog, or click the X in the far left of the palette.

This is what I got from pressing the "X" for fill:

Looks as I would expect for super thin strokes.

The status bar indicates the strokes are possibly a fraction of a pixel. They can be made wider, for visual effect.

If the document is for CNC, it might be ok to just view the paths without a visible stroke: Menu: View>Display mode>Outline.

So how could I increase the stroke width so that the image is actually visible?

I suggest exploring the interactive Tutorials in Inkscape's Help menu. Lotta good info there.

For greater details, there is the manual: http://tavmjong.free.fr/INKSCAPE/MANUAL/html/Attributes-Fill-Stroke.html