The picture depicts a speaker protection grille which needs to have the cloth replaced. I managed to source an apparently identical piece of fabric (a cotton/elastane mix according to the seller) but I experimented a bit and as expected I wasn't able to eliminate the folds at the corners on the back of the wooden frame. The manufacturer of the speakers apparently managed to do so and I was curious what technique is used.
This is a guess, but I think you simply have to use a glue that has a somewhat long working time (30 - 60 minutes).
After applying the glue, attach the grille cloth as you would stretch canvas unto a frame, from one corner to the opposite, and then attach the edges rotationally (the texture of the fabric in your photo shows this torque around the corner).
Once the mesh is in place, you can adjust its position to your liking. The excess fabric can act as a guide here to spread the fabric evenly, I imagine. Afterwards simply use a cutter knife to neatly cut off the excess.
The manufacturer of the original grille eliminated the folds by cutting them away. If you look very closely at the corner, you'll see that the feint lines of the fabric stop abruptly. That's where the excess was cut away.
It looks like the original fabric was glued on with a liquid glue. The process usually is:
- Apply glue around the edge and wait according to product instructions until the glue gets tacky.
- Place the new fabric on a flat surface.
- Place the frame on top, glued backside up.
- Fold one long edge over the frame and stick it to the glue, but leave the corner lose.
- Stretch the fabric on the opposite long edge and stick it to the glue, but leave the corner lose. This can be done in short sections.
- Repeat on the short edges.
- Now attach the corners: simultaneously grip the fabric on both sides of the corner and stretch it. You'll get a fold right down the center of the corner. Attach the stretched fabric and either push the central fold neatly together (standing up) it push it flat against the frame.
- With a very sharp knife, scalpel or razor blade cut away the excess fabric. If the corner fold stands up, cut it away with 1 - 2 millimeters excess, then push the excess flat against the frame. If the fold is already flat, cut as if it wasn't even there.
One example of the process can be seen in this youtube video. There are many more videos like this around, so if you're unsure it's best to keep watching some people do it and learn their technique.