Why do people think that going afk will fix the problem

Because they can do something else than eat Nidalee spears. Eating Nidalee spears is not fun.

I think that at least some players do it to avoid embarrassment, kind of like a "if i don't try, I cant fail" mentality. No one likes to be made to look bad, so when they start feeding or afk they are just trying to remove their own responsibility from how the match has progressed. But that's just my opinion

Because people are selfish and stubborn. They would rather punish a perceived slight against them, instead of sucking it up being a mature adult.

They don't. For them, the game is as good as over and rather than waste time on it, their mentality is that they should spend their time doing something else instead.

Basically, they don't give a crap that there are other real humans playing with them (to them, it is just PvE with really good bots), so they act like they would in any other PvE game. Some people are just so socially detached that they don't comprehend that they are giving someone else a worse time than they were having.