Probably, you created the image by vectorizing a bitmap. Internally, Inkscape creates filled shapes from the outer to the inner, and by breaking them apart, they all get the same color.
If you mark all as in your second step, you can ungroup them. Then you can move filled shapes from top to bottom to see, how they are arranged. In my case, the inner most circlesection is on top, then the second one, then the arrow, then the inner circle and last the outer circle.
I made a 3d-sheme of the order in 50% transparency, which is another way to deal with them, to gain your wanted result.
Depending on your needs, as already written in the comments, you can colorize the second circle in white or subtract it from the outer circle, or combine them.