It's used to evolve Togetic into Togekiss. This evolution requires a Togetic holding the Silver Feather and being traded to another player.
You can get the Silver Feather by trading it with another player who has obtained it in their game. It's not found in the wild or obtainable in the main game.
This makes the Silver Feather a valuable and sought-after item among players trying to complete their Pokedex and evolve their Togetic.