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Game starts out in the laning phase to maximize gold and xp gains. Even poor matchups are going to make it past laning(I'm aware it depends upon the game itself, I'm not sure when laning tends to end) and into the teamfight, so every team needs to build around the teamfight. Though because bad performance during laning gives a large burden to overcome it is also important to at very least perform adequately during laning.

For teamfights AoE, whether it be stuns or damage, is important. Not necessary for every party member to have, but very necessary for the team itself. Having both AD and AP means that one's opponents cannot just build armor or magic resist to reduce damage. Having a "tanky" character with hard CC to engage a fight means that either the other team will sacrifice one for an easy and non-taxing kill or force the other team to engage when the only easy target is the tanky character. Additionally for the damage dealers being a ranged character means that they can still act while being less likely to put themselves in harms way.

As such we have 3 characters determined, a tanky, an AP, and an AD. Since there are 4 places of gp and xp (3 lanes + jungle), sending someone to jungle only increases the team's gp and xp. This also means that one character is largely not seen on the map and could theoretically engage at any jungle opening, which means that opposing teams cannot push as hard as they could if all character locations are known.

We now have 4 characters. For the last slot, lets look again at gp and xp. there are 4 means of acquiring this, yet 5 slots. Any action upon the last character is not going to increase team xp and instead cut into one character's gp/xp. Engaging as a team 4v5 is just a bad idea, having the other character there should be a pretty obvious decision. Since xp is determined only by being near deaths, xp is always going to be cut into and that has already been decided. Since gp is determined only by last hits, this character could give up gp to whichever other characters are present. As the item progression in this game favors the later items (in terms of cost per stat points), the second amount of gold spent is more beneficial than the first amount or that giving up gold such that another character can have the gold is beneficial (only available in one situation since there are 4 revenue sources and 5 characters).

Now we have laning. There are four places and five characters. Obviously the jungler has to be in the jungle. Since the bottom lane is closest to the dragon and the dragon is the earliest point of interest to pop up (other than the lane pushes which have existed from the start), it makes more sense to send the second character to the bottom to secure the bottom and/or prevent the enemy from securing the bottom. And obviously the support has to be that second character.

This leaves me to wonder why exactly the rest of the placement is the standard is to have the bruiser top, AP mid, and AD bottom. Of course the top and the mid should have more sustain to prevent basing and keep up xp/gp gain and deny free farming.