How to repair damaged fence to its original strenght
There's more than one way to join galvanized wire, and a few ways you can do this. Some are better than others:

Solder: you can use solder to join ends. This is cheap if you have the equipment, and can do it in the field
Cable crimps: you crimp pieces of softer metal around the cable(s) you are trying to join. This requires crimps and a crimping tool, you can sometimes get away with using a pliers but the right tool makes a reliable crimp
Saddle bolts or cable clamps: saddle bolts are threaded u shaped bolts with a metal 'saddle', the cable(s) are put between the u and the saddle, which is then clamped down using nut on the bolt's threads. Cable clamps use the same principle, just a different shape. These are pretty cheap and you can put them on using basic hand tools with no special equipment. This is the option I would use, I would use at least 2, 1 isn't always good enough

Now as for how to do it, you have 2 options:

Butt end connection: this is where you overlap the 2 ends and then fasten them together using one of the methods above. This method is quick, has less joins and needs less slack, however I would not recommend it in your case as it's weak. Your wire is under tension, and the force will always be pulling directly at the join
Looping the ends: each end is looped back on itself and clamped to itself, the wires are joined through the loops. This is strong as the loops bear the load, not the joins. If you don't have enough slack you can use a loop of wire or a tensioner to bridge the gap. This is the method I would use, it may need more joins but it will last