How did the Resistance manage to win the war and destroy Skynet in the future?

Terminator

John Connor and his allies liberated a considerable number of people from the labour camps, forming the nucleus of his resistance army. They destroyed Skynet's command and control facilities and "defence net", then proceeded to destroy the mainframes that controlled the local defence systems around its primary location.

Notably, John Connor, using his superior technical skills (allied with his military prowess) was able to use Skynet's own satellites and defence grid to coordinate anti-Skynet activities around the world.

“There came a man ... a great man,” he added rever-ently, “who kept us alive. Ragged and half starving but alive. We got stronger, and he taught us to fight. To storm the wire of the camps. To smash those metal motherfuckers into junk. He turned it around and brought us back from the brink.”

_{^{Terminator: Official Novelisation}}

and

“It [Skynet] had no choice,” he was saying. “The defense grid was smashed. We’d blown the main frames—we’d won.

_{^{Terminator: Official Novelisation}}

and

Now, surrounded by his staff, John was coordinating a dozen major offensives throughout the world via his mobile telecommunications unit.

Reese had been told that they actually pirated channels off Skynet’s own satellites, knowing the enemy would destroy anything men could put up there but that it couldn’t afford to destroy its own global relay system. Reese didn’t have a clue how that stuff worked, but that wasn’t his job, anyway.

_{^{Terminator: Official Novelisation}}

Terminator 2: Judgement Day

The clear implication is that the human survivors, rallied by John Connor, were able to fight a war of attrition against Skynet's forces. Despite an initial advantage (the nuclear destruction of nearly 75% of the human race) and the creation of slave-run and machine-automated factories churning out Terminators, the humans are just plain outbreeding Skynet's forces as well as making it fight bitterly for every inch of ground.

To Skynet, human stubbornness made no sense. They fought when logic told the synthetic intelligence they were beaten. They relentlessly poured out of the rubble like a bacterial plague, their patterns of counterattack clever and difficult to predict. And humans reproduced at an alarming rate, their sexual appetites evidently fanned high by the threat of total annihilation.

Even though it took at least eight years before the human young could be made ready for battle, they were beginning to outpace Skynet’s manufacturing capabilities. And they were quickly learning to find the soft spots in the metal vanguard, decimating Skynet’s army of killing machines. Soon, there would be more human soldiers than nonhuman. The hyper-computer had miscalculated gravely on something it was still furiously analyzing: human will. So far, it had not come to a conclusion. And the war was grinding into its thirty-first year....

_{^{Terminator 2: Judgement Day - Official Novelisation}}

T3: Rise of the Machines

Skynet is actually holding its own in this timeline. Humanity is deeply disheartened and birth-rates are declining very rapidly. The inability to grow crops and supply food is taking its toll and John Connor and his forces have largely had to resort to hail-mary attacks on Skynet's headquarters. When a couple of these come very close to working, Skynet decides that now would be an excellent time to send another Terminator back to kill John Connor's lieutenants.

Human settlements, sometimes hanging on only by sheer determination, were critically important because they were the last pockets of resistance. They also were important because of the sharp decrease in the human birth rate. Who wanted to bring a child into a world of chaos, death, and destruction? These days the sparks of human existence were reduced to dim flickers around the world.

_{^{T3: Rise of the Machines}}

and

Skynet came to the same conclusion as John Connor. Something would have to be sent back. This second incursion on Navajo Mountain Redoubt had come dangerously close to succeeding.

_{^{T3: Rise of the Machines}}

Terminator: The Sarah Connor Chronicles

There aren't a lot of details about how the war is going in TSCC. We know that John Connor has invested a considerable amount of time and effort in developing and managing the time displacement equipment, to the point that he and his "Bubble-Techs" are able to use to send temporal hit-squads, covert support teams and reprogrammed Terminators through time with relative ease.

It's also apparent that he and his techs were reprogramming a lot of different Terminator models and giving them command roles throughout the Resistance. On top of that, he was trying to make contact with advanced Terminator models in the hope of coming to peace with them and securing their support against Skynet.

Garvin: Connor needs it.

Dietze: Connor? Connor is so into his big chess match with Skynet he doesn't see how the tin cans got us right where they want us. Metal on every base, running the show in all but name. Just waiting to hit us all at once with something big. Maybe something that's in that box.

_{^{Terminator: The Sarah Connor Chronicles - The Last Voyage of the Jimmy Carter}}

Terminator: Genisys

This timeline seems to most closely resemble the one seen in Terminator 2, with John liberating prisoners and leading the charge against Skynet. There are, however, two highly notable changes.

1) Kyle Reese is rescued by John much earlier in this timeline, during Kyle's childhood. It also seems that it's possible for someone to have two entirely different sets of memories after a timeline collapses.

2) It's explictly stated that John is able to predict the outcome of the war (along with the results of individual battles) because of the foreknowledge given to his mother in the T2 timeline by the T-800

Kyle: People whisper about John. Wonder how he can know the things he does. They use words like 'prophet'

At this point, both John and Skynet appear to be playing a bafflingly complicated game of 4-dimensional checkers, flinging soldiers backwards and forward in time to try to kill each other and to defend themselves. They're each using knowledge gained from earlier timelines to swing the battle in their favour or advance technologically.