The meta is merely the most optimal strategy to victory. It is not the only strategy to victory, and is_ not always universally superior_ in all situations. But it is effective for ensuring that 5 champions reach mid/lategame in an optimal manner.
solo mid- the mid lane has the shortest lane (in length/distance) meaning that they hit level 6 the fastest. Even more important; a champ at midlane can roam the the adjacent bot/top lane for ganks (which is much better than say, top lane which has to walk through the whole map to gank bot). As a result, the mid lane is kept solo, for roamers and champions that depend on early level 6s. (Tip: if you are mid, you should definitely be roaming to the sidelanes and ganking, especially once you hit 6 and get your ultimate; roaming is one of the big reasons why you go midlane)
jungler- the jungle is a viable source of income/exp, and by having one, it means that one of the lanes does not have to split up last hits/exp. Jungle also serves to provide global pressure primarily through ganks due to their lack of lane (which makes it difficult to track them, since they are in their jungle). (PS if you are a laner, yes, the jungler sacrifices lane farm so YOU don't have to compete for CS, so you better CS as best as you can.)
solo toplane- the top lane is a solo lane because there is no dragon at top lane. Dragon is a global objective that can be taken very early on in the game (usually as early as 5 minutes), because top lane has no global objective that impacts the entire team, there is no reason to send more than 1 champion, leaving it a solo-lane. Teleport is used to allow the top laner to provide global presence at bot lane. Usually because the lane is so far away from dragon, it rarely receives ganks/support so most traditional toplaners are tanky "meatwalls" that are hard to shove out of lane.
duo botlane- again, because the dragon is an important objective especially early game, having two champions botlane to help contest it helpful. The support + adc combo work well because the adc is not level dependant (remember adcs, scale with items and gold) meaning that they are okay with splitting exp (since they care more about autos than abilities), and the support is not gold dependant (or level dependant usually), meaning that they are okay with not getting last hits at botlane. (Pro-tip if you are support, the big reason why you and adc go botlane is because you guys can contest dragon, so make sure to ward it and fight for it!)