Oh, Kongquest's description of meta from a general perspective is great, but I'd like to contextualize why League's meta is the way it is.
What would you say is the most important stat in the game? I'll give you a hint, virtually every other stat in the game is in some way bound to this particular one I'm asking about. In short, it's gold. The team that both collects the most gold and then takes the best advantage of that gold is usually always the more successful team.
So how do you get gold? You can kill champions, you can take objectives, but at the end of the day, the vast majority of the gold you should be acquiring comes from farming, whether it's Minions or Monsters. Naturally you position your team around each potential stream of Gold: A Top Laner, a Mid Laner, a Bot Laner, a Jungler...and then the fifth player. We've run into a problem, what do you do with the fifth player?
We don't want to put him Jungle, as that impedes the Jungler's income. We could base him mid, and use his presence as a form of "roamer" between the lanes. In fact, this was the old meta; what we recognize as today's support could easily have been the "roaming" support of the team, focused entirely on chain ganking constantly.
Another possibility is to put the fifth player in a side lane. You lose the roaming potential of basing them mid/river, but you also gain a unique team building opportunity. See, side lanes are dangerous as they are geographically longer, and thus far more dangerous for a squishy carry to survive in normally. Think of how most Top Laners are almost always champions with built in tools to defend themselves, be it a plethora of CC, Tankiness, or some defensive gimmick like Riposte. If you base the fifth player bot however, and designate their entire job towards defending the bot laner, you can afford making the bot lane a carry, circumventing the danger of a long lane by giving the carry a personal bodyguard. This also makes amazing usage of gold due to how gold scales exponentially in this game: to rephrase, one fully fed carry and a gold starved support are faaaaaar more effective than two half fed non carries. Because AD Auto Attackers abuse the exponential effects of gold the absolute most, and because they are the least effected by the lack of experience due to sharing it with the fifth player, this creates the AD Carry-Support relationship we know today in bot lane.
Also, to answer the question of "why bot lane?", it's largely due to objectives. Specifically, Dragon. By basing the ADC-Support lane bot, you guarantee that there will tend to be more champions on the bottom side of the map and thus guarantee more support for taking on the Epic Monster in an emergency or at an opportunity. These are why AD Carries and Supports are synonymous with bot laners.
As an off note, we want a Tanky guy top to defend himself and we can afford a DPS bot because of the support. Since Mid is easier to make a carry lane due to its safer geography, we can put a Burst Carry mid. Assassins, Mages, and so on, this balances our team on the three major pillars of damage: Tankiness, DPS, and Burst. Tankiness is countered by DPS which is countered by Burst which is countered by Tankiness. By guaranteeing these three elements on our team due to how we designed the team based on the restrictions of the map, we maximize the team's potential at team fighting by making sure that there is a definite answer on our team no matter what part of the damage triangle the enemy team throws at us.
Jungle has its own meta partially independent from the laner meta, and the supports have their own triangle meta independent of the damage triangle that dictates our laners, but this is a comprehensive review of "standard" meta: Tanky top, Jungler, Burst Mid, DPS Bot, Support Bot. Like the effects of nature on evolution, it was outside pressures like the geography of the map and the need to maximize our gold that largely defined the metagame that champions perform most efficiently at.
