Build or buy the frame. You want a nice thick frame so that the egg crate foam will fit into it. Either use thick wood and framing tacks, purchase over sized frames or buy shadow boxes. Make a few larger panels (3'x4') and a few smaller ones (2'x2').
Cover the back of the wood with vinyl material and wrap it around the edges of the frame, tacking the material to the inside edge of the frame itself. Cut the material to make nice folds on the corner, and don't worry if the fabric gaps to show some frame. The important thing is to get a nice smooth edge of the frame and a back for the egg crate.
Lay the frame on the side covered in fabric and fill the inside of the frame or shadow box with the egg crate packaging, smooth side down. Be sure to get a snug fit to entirely fill the open space so that no piece would fall out if you were to pick up the frame and turn it over. Use super glue to keep the foam in place if necessary.
Cover the remaining open side with finer material, such as canvas. Make sure that the canvas is pulled tight across the frame and tack or staple the canvas to the back of the wooden frame itself. Do not compress the egg crate foam during this process, since you want it to retain its form inside the frame you've created.
Create art! Your soundproof tiles don't have to be bland and boring. Take the opportunity to create something new using the canvas as a painting surface. You could even replace the canvas with old t-shirts or pieces of a childhood bedspread to create a tile cover featuring your favorite childhood superhero!