Strip the ends of your wires about a ½ and inch and twist them together like in the first image, then cut it down just a bit before screwing your nut on nice and tight.
This is the inside of the splice. You can see the protective gel inside of it, this is what waterproofs your splice.
The nut with the wires goes into the splice, make sure you get it nice and snug in there. Some of the gel may come out around the edge, that is ok. Just snap it down and you are good to go. Your break is now repaired and you may now re-bury the wire and splice.
All finished. The repaired piece of fence is now buried back under the ground.
All prices are in USD