There are many ways to finish the ends on a rug hooking project, here's three of them:
1. Fold and sew
2. Whip with yarn/cording/taping
3. Whip with yarn
Perhaps the easiest way is to fold over the backing and stitch it closed. This is commonly used for pieces that are not overly large or will not be on the floor.
This was my second rug hooking piece, completed in 1998. The edges were cut about one inch outside of the pattern, folded, and stitched closed.This piece has been a display table piece and doesn't get a lot of traffic, and has aged well with this method.
After cutting the linen around the edge, use an iron to press the backing into place. This will make the stitching part much easier. If it's possible to get access to a serger (overlock machine), this can be an easy way to stop the backing from shredding before stitching it closed. To stitch the rug hooking project edge, the needle and thread can either be pushed back through the hooking itself, or can be stitched on the backing side only.
One of the main ways I previously would finish rugs was with cording, whipping the edges, and finishing with tape. The cord would be wrapped inside the edge of the finished piece. Yarn was whipped around the cording, which helps to create a nice rounded bump for the edge of the piece.
The taping is then stitched over the backing to hide it and protect the piece.Taping doesn't necessarily have to be used, but it does make the back of the piece look cleaner. The edges of the backing will fray over time, so unless a serger or a few lines with the sewing machine are run along the outside edge, it's recommended to cover the piece with the tape.
The third way is folding the backing onto itself a couple of times and whipping the edges with yarn to completely cover the backing. This option doesn't require tape or anything extra. While this doesn't make a bump like when using cording, it's hardly noticeable.
Finally, a less common (and trickier) method for finishing edges is using wool strips. This creates a small, tight border, and works well for decorative pieces where you're not looking for a lot of contrast on the edging.