Ok I'm gonna answer these questions in the order you presented them to me, so bear with me a minute while I go through this.
There's an enormous pool of champions that constantly fight in the Summoner's Rift, Crystal Scar, Twisted Treeline, and Howling Abyss. What are these places?
These places are locations in Runeterra with some kind of significance to the world as a whole, even if it's only a localised or historical thing. Summoner's Rift is a forest where once two forces battled each other ages ago. Twisted Treeline is a place on the Shadow isles and is the lair of Vilemaw; an undead spider god. The Howling Abyss was a bridge built by the Iceborn in the Freljord for their Frozen Watcher masters, and was the site of the last battle between those two forces when they went against each other. The battle is the reason the bridge is named as it is, because the Iceborn hurled the Watchers off the bridge and it is always described as "we tossed them howling in to the abyss".
The Crystal Scar's a bit weird right now because as you currently see it, it's not actually canon. The current canon right now is that the Crystal Scar is a location in the Shurima desert where the Brackern used to live. Within is the entrance to their underground domain where they are hibernating after being driven to the brink of extinction, with only Skarner remaining to stand watch over the land. I won't go in to what the Crystal Scar was in the old lore here but essentially the map isn't updated for the current lore about the location.
I know everyone has their own back story as to why they fight, but why was it started in the first place?
The fighting in game isn't canon to the lore, and neither is the gameplay. This is a distinction that was made about a year and a half ago now where Riot divorced the lore and the gameplay of League so they don't coincide anymore. So while all the champions are key figures from Runeterra and they can use all their in game abilities in the lore, the items are all artifacts and objects found in Runeterra, and the maps are all locations within the lore, the gameplay surrounding them isn't canon. To sum it up the game is effectively Smash Bros for the League setting.
Do they REALLY die in these places? In the newest cinematic, it depicts champions actually dying. Or do they actually respawn to fight again?
As I said, gameplay isn't tied to story anymore, so game mechanics and how they operate don't function in the lore. Champions die for real if they die in the lore, and bar magical ressurection (which is extremely hard to come by in Runeterra) then they're out of luck.
Right now League's in this weird transitioning period because Riot decided to do a reboot of the lore and removed a core element of it. To elaborate; when they game was released the reason as to why it existed in the lore was because an organisation existed called the Institute of War. They were a collection of magical Summoners from across Runeterra who joined together to stop magical warfare from going out of control and destroying the planet. To that end they created the League of Legends, where nations from across Runeterra would send representatives of their cause to fight in a small arena in the name of resolving political disputes.
Riot has been reworking the lore for years now and during the September of 2014 announced that they were removing the Institute of War from the canon, and lore would be adjusted around this. They did this because they thought that the contrivance of the Institute and the League was inhibiting their ability to create stories, for example wars were kinda impossible because the Institute could do away with them via the flick of a wrist, given how powerful the organisation was presented as, and some champions literally couldn't exist in the story without being prisoners of the Institute because they presented too much global risk to be let out, effectively shoving them in a box. There was a huge spat between Riot and the community on this and there's still some tension lingering.
This whole thing has left a gaping hole in the lore where some champions don't actually have stories anymore because their lore was so intrinsically tied to the Institute that they had no other functional lore about them to be salvaged for the new canon, meaning they exist in a lore limbo of just not having a place in the game's universe yet. Champions like Fiddlesticks, Jax and Nocturne don't have lore right now because what's written for them isn't canon. Certain parts of the old lore have been preserved, while others have been discarded, leaving the state of the old lore nebulous as we don't know how much of the former lore infers the new stuff.
Currently things are being worked on. There is no overarching story for the entire world itself and likely won't be unless there's some global threat. The world's history is being worked on extensively, as is the quality of the setting in general. Right now trying to explain how everything works in the lore is difficult because we as readers are actually missing a lot of the pieces for it, and they're being unveiled gradually. There have been big lore developments in the last few months though, and the future for lore looks promising.
Hopefully that made some modicum of sense, if not just say and I'll try and rewrite it in a better way.
