If the leather fits back in the clasp
It doesn't look like it in the picture, but if the leather can be inserted back in the clasp, I would use E6000 adhesive to glue it in. It bonds aggressively with both materials and remains flexible (shock-resistant).
Coat the end of the leather and get some glue over the tip and into the center of the braid. Give it a short time for some to absorb into the surface, then slowly push it into the clasp with a little back and forth twisting motion to spread the glue around inside the clasp. Wipe off any excess glue while it's still wet.
With the outside clean, wrap some tape around the leather and clasp to stabilize it so it doesn't move or pull apart while the glue is drying (painters tape or fresh masking tape should work without leaving residue).
I would give it at least overnight for the glue to dry (the clasp and tape will seal the joint and slow drying). Then carefully remove the tape without pulling on the joint. If you can still smell any of the solvent, give it at least another half a day to dry. Once you can no longer smell solvent, give it a day for the joint to reach full strength before using it.
If the leather doesn't fit back in the clasp
I'll suggest a few approaches based on the picture and some assumptions. With a lot of repairs of this nature, there isn't a standard way to do it. Basically, you want to connect the leather with the clasp in a way that looks like it did originally, or at least looks good and like it was intended to look that way, and so it stays together under use. I'll assume the method also needs to be doable without specialized tools you aren't likely to have.
The clasp looks flattened and there isn't obvious adhesive on the end of the leather. I'm guessing that the leather went into what started as a tube (the leather might have been compressed to reduce its diameter and inserted before it expanded again), and then the tube was compressed to make a tight friction fit. The magnet may have been glued in afterwards. The assumption was probably that there was enough friction holding in the leather that the magnets would separate before the leather moved.
Four repair approaches come to mind:
Secure the leather back inside the clasp.
With the clasp already compressed and a magnet in the other end, this would be hard to do without modifying the end of the leather. The key would be doing it in a way that's invisible.
Getting the leather into the clasp
You need to reduce the diameter of the end of the leather and make it stiff so that it slides into the tube fairly easily. Friction on the inside of the tube will cause the braid to expand, making it much harder to get the leather inside.
You can make a fixture to compress the end of the leather. Drill a hole in a material like wood that is a hair smaller than the hole in the clasp. Cut a thin slot through the wood on one side from the edge of the wood into the hole. Use a screwdriver in the slot to spread the hole a little bigger and get the end of the leather into the hole (no more than the depth of the hole in the clasp).
Remove the screwdriver and put the wood in a vise to compress the hole by pushing the slot closed. Leave the leather there for a while (maybe an hour). Then stretch the hole again with a screwdriver in the slot, remove the leather, and immediately push it into the clasp. If it expands too quickly, try it again with a fixture with a smaller hole or a wider slot.
If you can't compress it enough, remove a little of the leather from the inside of the braid. One way to do that is to secure the leather in the hole fixture, and carefully drill out a center hole to a depth of no more than the depth of the hole in the clasp (see note at the end of the answer). The idea is just to remove a little material and give the remaining material a place to go when you compress it.
Another thing that can help get the leather back into the clasp is to bevel the end of the leather with fine sandpaper. That might be easier if you secure and harden the end with something like superglue (cyanoacrylate, the low viscosity version; a very fine dispenser tip helps). Wrap a bit of polyethylene bag tightly around the end of the leather and then put a small amount of superglue on the end of the leather. It will soak in and also stiffen the leather at the end. In the US, most plastic grocery store bags and produce bags are made of HDPE. The heavy, translucent plastic bags inside boxes of cereal are also HDPE. Pre-test some superglue on the plastic to verify that you can peel it off.
Securing the leather in the clasp
If you can easily get the compressed leather back in the clasp and still have some working time, consider gluing it in as described at the beginning of the answer. If not, here's an alternate way.
Use a superglue dispenser with a "needle tip". Just outside the clasp, push the leather braid toward the clasp to loosen it, and stick the dispenser tip through a gap in the braid into the center of the braid inside the clasp. With the clasp down (so gravity is pulling the glue down into the clasp, inject the glue a little at a time into the center if the braid. I'm assuming the magnet has a good seal so the glue won't run out around the magnet. You want the glue to soak into the leather and run out the gaps in the braid to the inside of the clasp.
The glue will adhere the leather to the clasp and also harden the leather so it can't reduce diameter when it stretches when pulled to open the clasp. That should be plenty secure. If the glue soaks into the leather exposed outside the clasp, it shouldn't be very visible, especially if you wipe any free glue off the surface.
While you're gluing, keep some cotten swabs wetted with acetone handy to wipe off any excess glue (leather workers might want to weigh in on how damaging a quick wipe of acetone would be to leather, and any better way to clean off superglue, or post-wipe treatment). Don't use any form of dry wipe unless you want that permanently glued to the bracelet, turning it into a charm bracelet. :-) It would be worth doing dry runs with scrap materials to perfect your technique before doing it on the bracelet.
Secure the leather to the end of the clasp in a way that looks like it's inside.
The gist of this method is to glue some form of thin, strong, stiff but flexible cable inside the end of the leather braid. That is secured to a plug that fits snugly inside the clasp.
Something that could work for the cable would be cotton string soaked in diluted PVA glue and dried straight. Use something like a small diameter nail to expand the braid and open a hole down the center (it doesn't need to reach very far--1 cm is probably enough. Glue the cable inside the braid and stretch and work the braid tight around it. Many adhesives will work for this, but E6000 or Barge cement would be my first choices. Drill a hole in the plug, feed the cable through, and glue it in the hole (superglue via a needle tip dispenser), and to the inside face of the plug (E6000).
Tighten the braid so it's a slightly smaller diameter than the outside of the clasp, with a nicely squared edge. Glue inside the braid will hold that together. With the cable inside, you might need a fixture similar to what was described in the previous approach for compressing the end. Use that while the glue is drying.
Then insert and glue the plug inside the clasp (E6000), so the leather is tight against the end of the clasp. The gluing technique for the plug would be similar to gluing in the leather described at the start of the answer.
Secure the leather inside something that looks like it's an intended part of the bracelet design and fasten that to the clasp.
This approach is somewhat similar to the previous one, but instead of securing the fastening plug to the inside of the braid, it is secured to something decorative on the outside of the braid. For example, the end of the braid could be wrapped with jewelry wire that goes with the metal in the clasp. That would be wrapped very tightly, and would have the advantage over the original design of the wire ridges to grab the leather compared to the smooth surface inside the clasp. You could further use superglue inside the wrap to saturate the leather, hardening it so it can't compress and gluing it to the wire.
Secure the clasp inside the leather braid so only the other half of the clasp shows.
The gist of this approach is to loosen the braid at the end enough to create a hole that the clasp fits into. Drilling out a little of the internal leather would allow a thinner leather shell over the clasp. Glue the clasp inside the leather (something like E6000 adhesive or Barge cement would bond them together and hold the braid together at the end). Wrap the joint with tape to stabilize it while it's drying.
Note: Leather that's been hardened with superglue can be drilled out with regular drill bits. I've never tried to drill plain leather and don't know if a regular drill bit might chew it up, leading to damage visible on the outside. Plain leather might do better with some type of abrasive Dremel bit that sands away a hole. Any readers who work with leather are welcome to weigh in on this, or any of the rest of the suggestions. These are all extrapolated from having done similar kinds of things, but never exactly this kind of repair.
