You find the ends, pick them up, and there's a kit that can fuse the ends to your new piece. I'm not sure about the details, but I think there are repeaters on the wire that need to be added as well.
For submarine cables, there aren't that many strands, so getting it right is probably worth it, for terrestrial fiber, there may be 100+ strands, so they usually can sort things out at the ends, and I think you just fuse any strand to any strand. Sometimes, only some strands break and you can reallocate strands by priority until the cable can be fixed.
Strands in a bundle would likely have color coded sheaths. As long as you're not colorblind, it's pretty straightforward matching what gets spliced where.
Yes, commonly there are high-voltage cables in there.
Nowadays, for shorter runs (that are still long enough to need repeaters) there are also purely optical repeaters where instead separate fiber strands deliver light from a pump laser.
