What's going on with the backstory?
The game is based on the Legend of Drizzt series of novels by R.A. Salvatore, and the adventures of the same characters that you see/play in the game.
I'm not sure if there are other video games that cover these characters or this story, but to familiarize yourself with the setting and characters, you should read some or all of that collection of novels.
The Icewind Dale trilogy is the first set of books written and published by Salvatore, detailing the adventures of Drow rogue/ranger Drizzt Do'Urden as he strikes out on his own in The Sword Coast, a region in the famous Forgotten Realms world/lore. As the original trilogy, it probably does the best job at introducing the four playable characters of the game (and the world they inhabit).
Later on, Salvatore wrote a prequel series covering the origin story of Drizzt, called The Dark Elf Trilogy.
Here is the full list of books included in the overarching "Legend of Drizzt" series, in chronological order in-universe, with the first-published (and most-recommended) series in bold:
| Series | Titles |
|---|---|
| The Dark Elf Trilogy | Homeland, Exile, Sojourn |
| The Icewind Dale Trilogy | The Crystal Shard, Streams of Silver, The Halfling's Gem |
| Legacy of the Drow | The Legacy, Starless Night, Siege of Darkness, Passage to Dawn |
| Paths of Darkness | The Silent Blade, The Spine of the World, Sea of Swords |
| The Sellswords | Servant of the Shard, Promise of the Witch King, Road of the Patriarch |
| The Hunter's Blades Trilogy | The Thousand Orcs, The Lone Drow, The Two Swords |
| Transitions | The Orc King, The Pirate King, The Ghost King |
| Neverwinter Saga | Gauntlgrym, Neverwinter, Charon's Claw, The Last Threshold |
| The Sundering | The Companions |
| Companions Codex | Night of the Hunter, Rise of the King, Vengeance of the Iron Dwarf |
| Homecoming | Archmage, Maestro, Hero |
| Generations | Timeless, Boundless, Relentless |
(series information collected from Wikipedia)