Metagame of League is a pretty long story, but I'll try to make a tl;dr version at the bottom.
Basically, in a teamfight you want a frontline (to protect your carries), source of burst damage (to kill squishy carries), source of sustained damage (to kill tanks), and mixed damage (so it's harder to itemize against).
Burst damage champions are generally reliant on abilities rather than autoattacks to do damage. This means burst damage champions should go to mid lane, because that's the shortest lane and they will level up the fastest. In early-mid game, the mid laner should almost always be at the highest level. These champions are a main source of damage, so they need a lot of gold and experience.
Sustained damage champions can rely on abilities or autoattacks, but autoattacks are generally more reliable (no mana cost) and more sustained (attack speed is cheaper than CDR). These champions don't really need their abilities to do a lot of damage (generally). These are marksmen/ranged ADC. These champions need to maximize DPS (damage per second), so these champions also need a lot of gold, but experience is not as important.
Frontline champions need to buy tank items. Tank items are cheaper than damage items, so they don't need as much farm. Also, since they're tanky, during laning phase, their mistakes aren't as heavily punished.
Mixed damage is achieved by having a source of physical damage and magic damage. ADC does physical damage, and mid laners generally do magic damage (there are a lot of AD midlaners, so if that's the case top laner or jungler usually provide the magic damage in those cases).
Okay now, so why do different champions go to different lanes?
Burst damage champions (assassins and mages) go to mid lane because they need a lot of gold and experience. Mid lane is the best place because you can farm the minions and get the experience the quickest.
Sustained damage champions (ADC/marksmen) go to bottom lane because they need a lot of gold, but not so much experience. They go to bottom lane with a support because they can share experience. Support doesn't farm because they can be useful even without any gold.
Frontline (tank/fighter) go to top lane because it's one of the longer lanes. It's the most vulnerable lane to gank compared to mid lane, because it's longer. That means it's more difficult to run away from a bad fight. Since they're building tanky (generally), they can survive what could be otherwise dangerous ganks.
So why not have ADC+support in top lane and tank/fighter in bot lane? That's for dragon control. In early game, dragon is an important objective that can give you a serious advantage later. Having more people nearby will make sure you have an edge when you're contesting dragon. Baron doesn't spawn until 20 minute mark, at which point it's better to group up and teamfight.
There's a lot of exceptions to this rule, like Heimerdinger top. Heimer top is meant to harass the enemy top laner, since the enemy top laner is generally going to be melee champion. It's really difficult to fight a Heimerdinger as a melee champion because of his turrets.
tl;dr: mid laner needs gold and xp, so they go mid. top laner needs some farm but not as much as the carries, so they go to top lane alone. adc needs a lot of gold but not as much xp, so they go with a support who helps the ADC farm.