Top and Mid are solo because they level faster and get more farm.
Bot is duo so that you have 3 people + the jungler near Dragon for early game dragon fights.
A bruiser is top because top is a long lane, and you need to be able to stay in lane for a long time, and bruisers generally have better sustain than mid champions, and they counter AD carries. Sending a mid champ top would mean that have to go back more often than the other guy who sent a bruiser (in some cases) and sending an AD carry top would mean that the bruiser, most of which are at their strongest in the early game, would win the lane because AD carries are better late. This is of course subverted a lot, because the meta is more flexible than people think. Swain, Morgana, Mordekaiser, and Vladimir are all casters with enough sustain to go top and have strong enough early-mid games so that they're not at a severe disadvantage. Bruisers also often have escapes, like Jax's jumps, so they can escape ganks easier than casters, many of which have no escapes. Again, there are exceptions.
Casters go mid because mid is the shortest lane, so they can recall more often, and because casters often lack escapes, they go mid because it's the hardest to gank. Mid lane is also near blue buff on both sides of the map, so your jungler can hand blue buff off to the mid champion as often as possible.
Many AD carries have escapes, so going bot lane, which is as long as top, is logical, however, AD carries are all late-game champions, so you want to stick them with a support so they can farm easier. They don't need levels as fast as AP carries and bruisers, because their damage is largely from items and autoattacks, not abilities, they just need farm.
Supports go with AD carries because they don't need items to do well, which every other role needs. An Alistar can be extremely tanky with just a Heart of Gold, Merc Treads, and Aegis of the Legion. A Janna with no AP items still has a knock up, a strong slow, a very strong shield with a buff, and a knockback. Soraka is Soraka.
Junglers Jungle so that you can apply pressure to losing lanes to help them out, or decide an even lane, or even snowball a winning lane harder. Also you have two solo lanes so more exp and better farm for two lanes, and you have buff and dragon control. A team without a jungler is going to give the enemy team a free blue buff on both the enemy jungler and the enemy mid, an extremely farmed jungler due to having two red buffs and lots of extra camps, and if their lanes lose, they are on their own, and that jungler, once he hits the point where he can solo dragons, is going to have them for free whenever it spawns.
Kat isn't sent bot because she gets stronger with levels, and the AD carry (which you want on your team so you can win teamfights after 30 minutes) is going to have an extremely hard time against many mid champions, who get strong fast.
Now of course there are subversions to the meta. These are things that are in the meta, but are uncommonly done. Sometimes you send two bruisers to the bot lane to shut down a passive farm lane like Soraka/Vayne. Some people send their duo lane top to shut down a champ they dont want getting any farm, like Jax. However, this can backfire if it doesn't work. If the double bruisers don't get several kills, they're going to fall off, and fast, while only delaying the AD carry's super-strength (if that). and the 2v1 lane leaves someone on your lane in a 2v1 too, so if you don't win it harder than the enemy wins, you should have stuck with the meta.