A large portion of toxicity in League, in my opinion, is brought about by ignorance. League is a game with a huge learning curve and a lot of in game content to micro manage. A large part of the problem is, new players aren't really taught about these things. Most players just want to run around and kill sheet without thinking of the consequences of their actions. The game is about taking towers, yet in order to get a higher level mastery on a champion you have to have good K/D/A and Farm. The early stages of the new player process teaches nothing of positioning, lane freezing, when to roam, counter jungling, etc.
If I had a nickel for every time someone chased a kill into an unwarded jungle with no sign of enemies on the map and was punished for it, then flames their own team, I'd be a millionaire. Nearly every game I play people just want to chase kills instead of pushing waves and taking towers. Players also never think of their own teams win conditions. Most high elo players just say, "If you are good with a champ you can win any match!" Which is blatantly false, otherwise we would see more variety in LCS/Pro play. Players rarely think of counter picks or counter bans.
The game places the highest in game rewards on kills and assists, yet kills and assists are not required objectives to win the game. Towers are. I'm not saying don't kill an enemy champion, by all means if you are capable and the trade will come in your favor, kill them. However, most players think that if an enemy champion is low health that they can't let them get away no matter how far they chase, where if you don't chase, that low person, is either A) Going to go back to base and miss CS and Experience (a win for you!) or B) They will stay or come back low health, and you can just kill them then.
Toxicity comes into play when people don't realize these things and then get upset that another player didn't follow their stupidity.
Sometimes as ADC I won't feel comfortable going in or making a trade for various reasons, will it put me in the middle of an enemy minion wave, do they have a level advantage, do they have an item advantage, do they have hard cc, yet some supports I play with don't consider these things and will dive in. I will ping to fall back, then get flamed myself when they die and I live, even after I made them aware I wasn't going in.
Most people put themselves in the position that causes their toxicity.
The game does not do a good job of teaching the micromanaging that goes beyond K/D/A. In my opinion TOO much emphasis/reward is put on champion kills when it should come from objectives, specifically towers.
EDIT: Also, knowing when is a good idea to push a lane vs taking a dragon or getting baron. There are times when we could have won a game if we had pushed, but lost because some people thought it would be a good idea to do something else. When you have people that don't think ahead about the consequences or foresee an advantage, mistakes happen and cause people to become frustrated. There is a mindset also that if one person on your team is doing bad, you shouldn't help that person anymore. That is bad for your team. Why would you want someone to stay weak, risking becoming even weaker and being useless to your team. K/D/A promotes selfish play. That is why I think emphasis on that is toxic.