I first made these boxes several years ago for one of the Craft Challenges on Craftster. I didn't win, but I thought they were still pretty cool anyway.
1. Make a box out of popsicle sticks as shown below. I recommend using regular white glue to adhere the sticks, because it is easy to reposition them that way. Build up the sides of box until the depth is at least one inch.
2. Measure the depth and the length (or width) of the inner bottom of the box. Since the the box should be roughly square, you need only measure one side of the inner bottom of the box.
3. Using measurements taken in step 2, draw a square onto your felt as shown below. As you can probably tell, this doesn't have to be a super meticulous process. The goal is just to cut out a piece of felt an approximate size and shape that will let you drape it in the box to do further fitting, without wasting too much felt.
4. Cut into the corners of the felt about 3/4" to allow flexibility for draping the corners.
5. Place your felt in the box and smooth it to fit. If your felt piece is a little large and extends past the edges of the box, go ahead and trim it to fit at this time.
6. Cut small notches in the felt at each corner of the box. There should be 2 layers of felt overlapping each other at each corner, so be sure to cut a notch on each layer. I recommend notching each layer individually to minimize mistakes.
7. Remove felt and cut out corners as shown below, using the notches as guides.
8. Final felt shape should look something like this. As you can see, it is not perfectly exact, but that's okay as long as it completely covers the inside of your box. There's no reason to get crazy and spend a lot of time on getting this piece perfect. You won't be able to tell the difference once it is adhered to the box.
9. Paint your box and let it dry completely. I painted the whole thing so it would look pretty for the picture, but it's not really necessary to paint the parts that the felt will be covering.
10. Brush a layer of white glue all over the inside of the box, and then insert, position, and press your felt in to glue it into place.
11. Cut and prepare pieces to be glued into the box. Add pipe cleaners for support, as shown below, if desired. Glue your pieces into place using the hot glue gun, and also add any other desired decoration at this time.
12. All finished! Now you may admire your masterpiece.
More examples of finished shadow boxes: