The fact that everyone seems to think that Support follows the AD Carry around is a bit discouraging, then again its understandable considering the current meta has been around for a year.
For the longest time the AD Carry went mid & the AP Carry + Support went botlane, Before that you had double Bruiser bot.
Anywho, lets break down why each champion goes into said lane.
A) Mid Lane --- AP Carrys go into mid lane for a few reasons
1) They have massive power spikes with just levels alone, typically 6-11, Mid lane is the first for creeps to reach it meaning they gain exp faster than anyone else on the map. --- Because of said power spike putting them in the exact middle of the map allows for a wide range of uses, such as; Defending Jungler, Roaming, Pushing, Invading, Objective Control.
2) With the massive power spike, you want them to soak all that exp alone, however its dangerous to be a mage alone, which is why mid lane is perfect, its so short that you only need 1 person to cover it and be perfectlly protected from most danger.
B) Bot Lane --- Why AD/Support?
1) The AD Carry is the weakest role on the team in the early stages, typically they just want to farm up as much gold as possible, because they become late game power houses with full item builds.
2) Because of how weak they are, you want someone to be in-lane with them, this is how it went from Roaming Support to 0 CS Support, being able to protect and defend the AD Carry while they farm and still being useful without much gold means that only a select few champions can do this.
3) This became standard because it allowed Support to always keep up in levels, where you would typically fall behind if you failed to make several successful ganks, It also guaranteed that your AD Carry was going to become a power house as the game progressed onward to the later stages.
4) Why Bot and not Top? -- Dragon Control -- It spawns at 2:30 While Baron spawns at 15:00, Dragon can be done with 2-4 people quickly, allowing for early advantagesto the team.
C) Jungler --- Why?
1) With the use of Runes & Masteries, it became reasonable for many champions to farm all that untapped Gold & Exp in the Jungle, while still keeping up fairly evenly.
2) It opened up Top Lane to get double the Gold & Exp for becoming a Solo Lane
3) You gain massive Map Control with the ability to quickly get Red & Blue buffs at the start of the game, as well as the constant fear/pressure of being able to gank lanes.
4) Leashing is an extension fo this, in which you set your Jungler up with THE BEST POSSIBLE START, allowing for early ganks & counter jungling to put the enemey at a disadvantage.
D) Top Lane --- The Power Creep
1) Since this lane is just as long as Bot Lane but with only one person & the potential to be put in a 3v1 situation due to the Jungler & Roaming Mid. You needed to be very specialized to survive in this lane.
2) Champions that have multiable of the following: Innate Sustain, Defense, Escape, Dueling Prowess, Etc. Thrive in this harsh island expierence.
3) This created a design creep of Power & Mobaility that could only be beaten by others with even stronger 1v1 aspects, Thus leading to the very snowbally Top lane of today.