What it boils down to is this;
When a person dies, and they leave some unfinished business behind, but they were kind of a bad person, this high-ranking demon named Malebolgia--it isn't even actually Satan, or the devil, because that exists as a higher up--offers up the person a deal. In exchange for the dead person becoming a soldier in Malebolgia's army, he will allow them to temporarily go back to Earth to deal with their unfinished business.
But Malebolgia, being a demon, fucks them over pretty good. He now has their soul, and whatever business they had to attend to on Earth is usually fucked up beyond repair, so they really have not choice but to just go back to hell. Some of the Spawns, realizing they got fucked over, are none to happy with Malebolgia and defy him.
The main Spawn in the series is a guy named Al Simmons, who was like an Assassin/Mercinary/Military guy. Long story short, he got fucked over by his boss, and killed buy a guy under the order of his boss, so Al wanted to go back to Earth to sort things out and see his wife one last time. But Malebolgia fucked him over and sent him back to Earth five years in the future where he is nothing but a rotting corpse and his wife is remarried with a daughter, and his boss got away with his murder. Malebolgia is laughing down in Hell, but Spawn is constantly not using his powers, (once the powers are used up, he immediately goes back to hell) so Malebolgia is starting to get irritated. Spawn starts running a gauntlet of demons that Malebolgia throws at him, along with Angelic threats (since Hellspawns are the enemies of Angels) along with other crazy monsters and beings that are just drawn to the hellspawn powers.
That's how it worked early on in the comic. I know later on a bunch of stuff got changed, and retconned, but basically the reason why the Spawns fight back is because they got fucked over and are defying hell.