OK, if you want to rearrange the pieces (like in a jigsaw) then each triangle/polygon will have to appear in a rectangular bitmap with a transparent background, because that's how drawing bitmaps works in Java/Android (and most other environments).
There is a way to do this sort of masking in Android, its called porter-duff compositing. The Android documentation is poor to non-existent, but there are many articles on its use in Java.
Basically you create a rectangular transparent bitmap just large enough to hold your cut-out. Then you draw onto this bitmap a filled triangle (with transparency non-zero) representing the cut-out. It can be any colour you like. Then draw the cutout on top of the source image at the correct location using the Porter-Duff mode which copies the transparency data but not the RGB data. You will be left with your cutout against a transparent background.
This is much easier if you make the cutout bitmap the same size as the source image. I would recommend getting this working first. The downsides of this are twofold. Firstly you will be moving around large bitmaps to move around small cutouts, so the UI will be slower. Secondly you will use a lot of memory for bitmaps, and on some versions of Android you may well run out of memory.
But once you have it working for bitmaps the same size as the source image, it should be pretty straightforward to change it to work for smaller bitmaps. Most of your "mucking about" will be in finding and using the correct Porter-Duff mode. As there are only 16 of them, its no great effort to try them all and see what they do. And they may suggest other puzzle ideas.
I note your cutout sections are all polygons. With only a tiny amount of extra complexity, you could make them any shape you like, including looking like regular jigsaw pieces. To do this, use the Path class to define the shapes used for cutouts. The Path class works fine with Porter-Duff compositing, allowing cutouts of almost any shape you can imagine. I use this extensively in one of my apps.