Abbie's answers are great and should work! However I wanted to provide a couple other options.
Folding the edges under
It can work. When you cut out your letters, add a seam allowance to all edges. Clip the curves and cut the corners with nice sharp scissors, then fold under and stitch.
Clipped corners, done properly, shouldn't fray a lot. A crisp cut will give the fabric flexibility to fold under neatly, and you can roll it a tiny bit more to hide the clipped spot. However, you can always add a dot of fray check to the clipped points if you are worried about it.
Make a facing
The preparation is similar to folding the edges under, except you are attaching a separate piece of fabric to the edges and folding that under instead. To do this:
- Before cutting, add seam allowances to the edges of the letters
- Do the cutouts along the edges (with seam allowance added), both on your visible layer of fabric, and on the facing fabric (will end up on the back)
- Sew the facing to the front along the cutouts, right sides together
- Clip facing, turn, press; optionally topstitch along openings
You can finish the edges of the facing fabric however you like since they won't be visible. However, this technique wouldn't work well with sheer or thin fabrics, as the facing could add visible bulk. This will also mean the finished piece is not reversible.
Bind the edges
Get or make some binding, depending on if you want it to match or contrast. Woven binding tape is cut on the bias, which allows it to stretch a little around curves. However, on corners you will either have to make miters or folds. It can be tricky to apply properly, so you may want to practice first if you haven't worked with binding before. This will add a little bulk on the edges and (if you are careful with your stitching!) can result in a reversible piece.