The Medallions have no effect on gameplay what-so-ever. They're a plot device. In the same way the Crystals and Pendants were in A Link to the Past or the Triforce Shards were in the original TLoZ, they act as little more than a way to keep score and track your progress through the game on the quest status screen. They are indications of your victories. Trophies.
Contextually, IDK where they come from. They may simply be an emblem to represent their respective sage. When it is stated that "The Sages will add their power to yours" that is a reference to the sealing of Gannon at the end and the sages using their power to provide Link access to his tower. All six sages, plus Link and Zelda (as the seventh) had to work together to seal Gannondorf. This is all story, it has nothing to do with gameplay.
Although, the hacks that have been done on Ocarina of Time have revealed evidence that suggests that early on at least one of the medallions may have been intended to cast some sort of spell, that may have later evolved into one of the three spells Link acquires in his adventure. IIRC it may have been the Wind Medallion, which of course does not appear in OoT, but there is also evidence to suggest there may have been plans for a Sage of Wind and a Wind Temple, that were scrapped.
