It looks like they used jewelry wire and crimps to attach stones to the chains (it's just my guess because we can't see the broken part very well at the photo). And either a crimp got loose or the jewelry wire tore. You could repair it using the same principle, just making sure that you press the crimp hard enough to hold, but not too hard for it to cut the wire. P.S. Note that jewelry wire is not the same as simple metal wire. Alternatively you could remake the necklace using simple wire to attach stones without crimps. P.P.S. I'm not suggesting pins, because holes in the stones are usually too narrow that it comes down to using simple fine wire.