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Both sleeve ends of a jacket end in this type of stitching, which after a decade, is starting to come apart. The jacket itself is Tyrolean, made of loden, which is a type of melton wool. I don't know what the yarn used for the hem is, but the thicker one is almost certainly a wool.
How could a novice sewer like me go about replicating and fixing this by hand?
The "Stitching" on the edge of your sleeve, is actually a strip of fabric, that was folded around the raw edge of the loden cloth, and sewn to the loden cloth.
This strip of fabric is called "Binding". (it's not piping, because there's no cord running through it. The puffiness comes from the thickness of the boiled wool loden cloth inside it).
The binding on your coat has a knit construction, making it flexible.
If I wanted to fix this coat by hand, I would find some yarn (worsted wool, cotton, or synthetic) that matches the binding (yarn comes in a great range of colors and weights) and sew it over the binding with a close-set blanket stitch. The blanket stitch creates a binding, because it wraps around the edge of the jacket. So it will cover the frayed part of the old binding. https://s-media-cache-ak0.pinimg.com/736x/0a/d3/91/0ad391d598c823cf864f51ff6f30edfc.jpg
Blanket stitch is easy to learn - find a youtube video. Use a heavy crewel embroidery needle.
Fiber choice - For durability, avoid rayon. Nylon and polyester are strongest. Choose yarn or binding that is smooth/firm/tight rather than fuzzy/soft/loose, for dirt and abrasion resistance. (cuff is subject to dirt and abrasion). Avoid eyelash, chenille, and other novelty yarns. Worsted (tightly twisted) wool sheds dirt.
Binding examples
Yarn examples
In order to answer, I want to back up a minute, and explain a bit about how loden fabric is made. Loden is a type of felted/fulled wool fabric, which means that an original fabric is created by weaving or knitting (per this website, loden fabric is woven). Once the fabric has been created, you basically get the fabric wet and then abuse it mercilessly until the wool yarn attaches itself to its neighboring yarns, and shrinks in size. This creates a dense, fairly waterproof, hard-wearing fabric.
This fabric is what is used for the body of your jacket--however, fulled/felted wool fabric is not particularly flexible and so instead of using the felted fabric to make the piping on the edges of your sleeves, the manufacturers used fabric made by knitting the yarn that matches the jacket color in stockinette stitch (they actually used two types of yarn when knitting, matching wool and some sort of embroidery silk or cotton, but I can't really tell what the other yarn type is from this picture).
The knitted fabric was then wrapped around a cord of some sort (it looks dark brown or black in the pictures), and then sewn closed to make piping. Imagine that the inner cord is a hot dog, and the knitted fabric is the hot dog bun. The knitted bun is wrapped around the hot dog and then fastened closed at the top so that you can no longer see the hot dog at all. This piping was then sewn onto the sleeve edges (actually, this is where the hot dog analogy breaks down, unless you can imagine a very long flexible hot dog that is bendy enough to make a circle the exact diameter of the cuffs of your jacket).
I think you will have a very difficult time finding materials that are an exact match to the sleeve that doesn't need repair, so I would recommend doing the same thing on both sleeves.
Basically, I would take a sharp pair of scissors, and cut the cuffs off of each sleeve. As long as you cut the felted fabric, the edge will remain nice and neat, because felt does not ravel. You can find some woolen fabric that you think will look good with the jacket, and create new piping for the sleeves (there are lots of good tutorials for creating piping--here's one). Then re-sew your piping to the cuffs of the sleeves.
You could also look for pre-made piping and sew that to the sleeve cuffs yourself. However, most pre-made piping is used for home decor, and might not hold up to laundering or look right with the jacket. You could look on a site like Etsy to find someone who makes piping out of a variety of fabrics--maybe you can find someone to make something to your specifications, so that you can use a fabric that is appropriate to your jacket.
There are a few other thoughts that I have about potential repairs, but I think this is going to be your best option for long term durability.
Edited to add: After reading @Robin L.'s response, I think she is correct that this is not a piped edge treatment, but simply knitted fabric wrapped over the edge of the sleeve. I think that her suggestion for blanket stitching with a matching yarn is a good option (and much simpler than mine). I actually have a boiled wool jacket myself that has sleeves finished with blanket stitch.
