How is the background removed from this image?

[Nidhogg]

Can't say how teapot extracted it. I haven't done any complicated background removals for awhile,
but for the spokes I can say that it's the easiest part if you do it separated.
That's because they are just straight lines. You could use path tool, merge all the spoke paths, stroke and then create a mask of it.
Wand, color selection or color to alpha do not work on image like this. I never don't use those tools for background extraction.
Usually I start with some of the luminosity script/plugin to get a version that has as much as possible contrast but still soft edges.
From that I build a b/w version that is used as layer mask and then I apply the layer mask on the image.

[teapot]

The cutting out left lots of missing spokes. I drew them back in with the paintbrush tool. (Easy to get straight lines, left mouse button click, hold shift key down, move mouse, click again).

[Wallace]

This was done with a layer mask.
I bit time consuming,
but very effective if enough attention to detail is given to the template.
You could also simply paint onto the layer mask itself,
instead of creating a template for the mask.