There are a few ways to do that.

With the selection tool:

• First select your subject as usual (around the "clear image" part)
• Enlarge your selection by half the width of your "fuzzy border" with Select->Grow (so if you want, let's say 50 pixels of fuzzy, grow the selection of 25 pixels)
• Smooth the selection with Select->Feather for the same amount of pixels
• Copy the selection; You should have in the clipboard what you're looking for

With a layer mask:

• First select your subject as usual (around the "clear image" part)
• Right click on your layer and select "Add layer mask"
• Initialize layer mask with "Selection"

A layer mask is basically a grayscale layer where color will be copied into the alpha layer of the target layer (white=fully visible (100%), black=invisible(0%)). So you may now use any filter you like to add a black to white gradient border to your white object of interest; when you like the result, right click on the layer again and select "Apply layer mask" (In your case, a blur filter should be a good start)

Alternatively, you may "paint" your selection using regular painting tools directly on the layer mask (Use white to extend the selection, black to reduce it and any other gray for alpha effects.

There are probably other ways to do that, like using the "Toggle quick mask" button, but they are mostly variants of those two solutions.

Bottom-line: The layer mask approach is ways more flexible, and worth spending some time learning how to use it if you are considering using Gimp seriously, but in your case, the first solution is probably good enough, and will be a lot faster if you never used layer masks before.