The AD carry tends to be weak early, but scale incredibly well. Having an adc on your team means having better neutral objective slaying, as well as tower killing. As adc's tend to be more item reliant than gold reliant, it doesn't matter if they are solo lane or duo lane; they just need a high level of income.
As a support, you have a lot of decent base stats or raw utility that does not require a high level or high level of gold to perform. Obviously both of those allow you to support better, but you can get away by being much lower level than the rest of the players and still be impactful (look at pro matches, you regularly see supports 3 levels behind the curve).
Because these are two roles that don't need a lot of exp, and there are only 4 reliable sources of exp (top, mid, jungle, and bot) putting the two less xp dependent roles just works out nicely. Also because support does not need as much gold, it allows the adc to get all the farm.
The reason they are bot lane specifically is because the dragon used to give team wide gold to the entire team. Having extra people bot lane meant it was easier to try to take the dragon after a successful gank or lane shove. If we get to a point that rift herald is stronger than the dragons, it would not be too far fetched for the duo lane to move top side.