I just don't understand why x should go top and x should go bottom. Or why top should be solo and bottom should be 2. Who made up this metagame?
Each lane has different attributes which makes them condusive to different champs.
These are:
1. The size of the lane (distance from tower to tower). Top and bottom are about the same distance, whereas mid is signifigantly shorter.
2. The time it takes to return to lane when you leave. Top and bottom take about the same time.
3. How close it is to other lanes. Mid can move to either lane with little issue.
4. The neutral objectives nearbye. Dragon is near bottom lane whereas baron nasher is near top. Dragon can be taken at almost any time wheras baron can't be taken till the endgame when lanes are ignored.
Top and bottom could basically be exchanged if it weren't for dragon. Since you can start taking dragon at level 4 with a large enough advantage, having two people there is a very significant advantage when dragon fights occur in terms of control, especially when one person is free to ward.
This means that top is relatively isolated from the team at large because it's not as important, so you need somebody that can stay in lane for the long term or bully the opposition early, hence you have bruisers and bruiser mages there, they need to be able to stay in lane without support, especially since most people there have high base stats and melee creeps. Plus getting back to tower takes forever if you're overextended and returning to lane is a massive time sink. Generally bruisers scale best with gold, but also scale well with xp. Mage bruisers scale like your general APs. This lane works for both.
APs scale best with experience because the base damage of their spells is so high and they don't gain anywhere near as much as AD carries do from gold, while they want farm too, it's more important that they get the earliest creep waves and therefore have levels earlier then anyone else. They're also strongest midgame because of this.
They also generally have the ability to clear creep waves very quickly which means they can be free to gank other lanes without losing farm or xp. They're also very squishy, so having only a short distance to run back to turret helps a lot.
The Ad carry scales best with gold, but not that well with xp, so they don't mind giving up xp to lane with somebody else. Supports scale primarily with xp if at all, so they can afford to spend the majority of their money warding instead of farming, and spend their time bullying the enemy lane so their AD carry can farm. The support can also keep dragon warded, which is equivalent to multiple kills.
Jungle can really be any in teamfights, the point is not to waste farm essentially. In practice, bruisers tend to be best at jungling though, because in addition to not wasting farm, they can pressure the enemy jungler and pressure the opposing laners which can allow your laners to get an advantage they can use to deprive the enemy of subsequent farm, creating a snowball effect.
TL;DR: So, this setup offers:
1. Every lane is optimized in their farm.
2. neutral objectives are controled during the laning phase.
3. The lane that is most able to gank other lanes is in the best position to gank.
It basically offers everything you could want.
Note: This doesn't mean that you should always stick to the meta, just understand what it offers and be sure that either you can force a win in 20 minutes or you have a way to provide everything you were trying to get so your team still scales into teamfight phase. This is basically:
1. Some bulk in your team
2. Initiation (can be on any role), it's primarily crowd control.
3. Magic damage, preferably with signifigant aoe but instagib also works, as long as your burst has a low enough cd to use it multiple times in a teamfight. This is primarily to soften their their team up.
4. Somebody that can deal consistent damage in the form of auto-attacks.
5. Pushing power so you can push waves to turrets.
6. A divide between physical and magic damage so they can't just build mr or armor to destroy you.
An example of a counter-meta way to achieve an AD carry is a roaming/jungling twitch, but he NEEDS to get fed otherwise he's far behind. 6 item twitch is still the most powerful AD carry in the game though.
Also, keep in mind some unconventional roles work.