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.
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.
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.