Why the Hexagon?
If there is one thing that best represents the Old School role playing movement/revolution/renaissance/zeitgeist/community/thing it would be the hexagon. D&D used hexes. So did Traveller. Role playing came out of wargaming and so it was the defacto way of representing the wilderness soldiers, explorers, and characters wandered through. All the early D&D campaigns used them: Greyhawk, The Known World, Blackmoor, and The Wilderlands to name a few. We can clearly say nothing says "Old School" like "hexagon" because "hexagon" says "war game." They work well for maps and breaking up an area to keep track of where stuff is, be it deep space or high mountains.
Where did it come from? Where did it go?
Pioneered for gaming by the RAND Corporation, the hexagon was picked up by Avalon Hill and found its way into the early days of role playing. Hexagons made gaming easier and made movement through trackless wilderness track able. My personal belief is that the removal of Demons and Devils from the game was not a reaction to the religious criticism of the era but rather an attempt to move D&D away from its wargame roots towards the storyteller railroad paradigm it had been drifting towards since the mid 80's. Get rid of Demons, Devils, Sandbox settings, "challenge the player", player-DM cooperative world development and hexagons and you could replace them with proficiencies/skills, railroading adventures, DM PCs, laundry quests imparted by Elminster, and sweet sweet pretentious drama.
What are their uses?
Hexagons are extremely useful for gaming. The right size of hex is great for judging distance. If you know the distance from face to face you can figure out the distance of each edge, the distance from the center and the distance from point to point. This allows you to figure out distance to a fairly decent ball park, especially if a DM uses a hex in hex system where hexes on one scale can be represented by hexes on another. The same system hexes in hex allows for cataloging the game world. If they are numbered a DM can keep a record of what is in each hex. This allows the DM to streamline his resources to develop only the places the players are going to go. The DM only has to provide detail when it is absolutely necessary but the players are not limited to a set path. If you use center to face and center to point you have twelve degrees of movement, not just six. Hexes help to generalize terrain. There are numerous other more subtle uses of the Hexagon.