Why are the Heroes of the Storm fighting?

The world of Heroes of the Storm -- that is, all of the various realms in which the battles take place -- is collectively known as the Nexus, a vast interdimensional storm.

Most of what we know about the Nexus is explained by Uther to Raynor during the tutorial: it periodically "grabs" heroes from other realms, including Azeroth, the Koprulu Sector, and Sanctuary, transporting them into the Nexus to do battle in its various realms, and "in the Nexus, death is only a setback".

In-universe, there is no explanation for why the Nexus chooses particular heroes. Even Uther, in the tutorial, admonishes Raynor not to think about it too hard.

The rulers of the various realms within the Nexus always serve as the announcers for the battlefields that they control and are presumably some kind of demi-deific powers. The closest thing we get to a justification for why anyone is fighting at all is that the announcers seem to enjoy watching -- or participating in -- the battles just for fun:

Captain Blackheart: "What? They're all paid up? Bah, let's get to the fun part!"

Queen Nightshade: "YES! MARCH FORTH! UPROOT YOUR ENEMIES' FORTS!"

Some maps are also contested, with separate announcers for each side (the Battlefield of Eternity maps and Towers of Doom), so the Nexus seems to have its own conquest-driven internal politics too, but mostly it seems that the rulers like watching the wanton destruction and brawling.

The realms of the Nexus include:

• the Battlefield of Eternity, pulled in wholesale from Sanctuary, or possibly Pandemonium
• King's Crest, a medieval world that includes Blackheart's Bay, Dragon Shire, and the Garden of Terror
• Raven Court, a gothic horror realm that also includes the Haunted Mines
• the Towers of Doom, a new realm on the edge of Raven Court for which the Raven Lord and Grave Keeper are battling
• Luxoria, an Egyptian-themed world that includes Sky Temple and Tomb of the Spider Queen

Some of these realms have had brief hints at their history and mythology through skin releases; for instance, Valla, Kerrigan, Nazeebo, and Arthas have all had Raven Court-themed skins, chronicling the saga of the bloodthirsty Countess and the vampire slayers who hunted her; and Tassadar, Zagara, Brightwing, and Xul have Luxoria-themed skins. Many of the other skins have been implied to be from elsewhere within the Nexus as well, such as Star Princess Li-Ming and Space Lord Leoric.

Blizzard has stated that the Nexus is absolutely not canon within the Warcraft, StarCraft, Diablo, and Overwatch franchises, but does have its own internal canon.

Incidentally, Thrall, voiced by Blizzard's Creative Director Chris Metzen, has a few things to say about all this if you poke him enough:

What do you mean there's lore in this game? You guys actually paid someone to write a story about Raynor meeting Diablo? Isn't this PRECISELY what fan fiction is for?! ... I didn't approve any of this!