Allright, so normally the animated GIF's I have seen are usually fairly small, and each frame does match exactly the size of the first frame and is filled with the pattern to be shown.

The example picture of a dog is a bit different. There is one background layer with all content, and on top of that 60 layers with transparency set except for a small area that contains the dog's head and front paws. The aperture is same on all the top layers, so the animation effect is only contained on one small area.
This is really efficient way to keep the file size small compared to storing and redrawing the whole frame.

I would guess this is achieved by taking normal short movie sequence, then analysing the subsequent frames for movement/change that is over some treshold (we don't want tree leaves moving in wind I guess) with keeping precedence on the moving bits in the center of the frame, on the assumption that the user wants to emphasize the object near the center.

Then, calculate a mask for just the moving parts (mask could be single aperture an on this exampe or multi-aperture if there are more than one loci of movement)
I assume that the mask has to spread the area of movement through the entire sequence, so that it is identical for all frames. Otherwice the redraws would propably look quite messy.

This mask is then used to cut out the differing parts in frameset, and finally written to a single GIF file.

Adding this:
In the Lumia you actually take a film sequence, then you get the possibility to "paint" with your fingers on the screen what parts you want animated or (if you click on an icon) not animated (static).

Ok, this makes the whole thing simpler as there is no need to calculate the aperture mask.
I'd say it is fairly simple to implemet using imagemagic libraries.

There are also instagram like effects and visual indicator of the frame being played that you can manipulate to make the GIF smooth.

And yes you can indeed paint what parts are visible and not, so you can make GIFs like below.