At the surface-level, the Blades influence play with their appearance. They are strange, menacing, often partially organic, and kind of creepy. Some of the strongly-themed Blades provide hints as to their origins and personalities, such as the archangel feather Michael or the Japanese demon Oni — although the entities themselves are likely to be simply co-opting mythical symbology to influence their wielders. The Blade designs act as a visual cue that the players can riff off and use to expand their own backstory and personality. The players might also decide that their Blades communicate telepathically, or even that they can speak, but that's down to freeform roleplaying.
The Techniques unique to each Blade also influence the way the Chosen fight, opening up different approaches and playstyles.
Blades deeply influence the core of the system through the Power they provide to their wielders. The Blades want you to have more Power than you can control, because then they can take you over and pursue their true objective. To that end, they tempt you to increase your Power each time you fight. It's your choice... but it's so tempting, and if you don't take the opportunity you may find yourself quickly outmatched. And if you lose too many fights, you'll find your Will eroded, and the Blade will start to use your moments of weakness to involuntarily feed you more Power. Once you gain Power, you can't decrease it — so every point ramps up the pressure to keep your Will high.
It's when a Chosen finally overreaches and becomes Bladebound that the voice of the Blade is most clearly heard. The character loses their free will, and must follow a prioritised list of actions. The Blade ultimately wants its Enmity (one of the other Blades, against which it holds an age-old hatred) to become Bladebound so the two can duel in their true forms, and one will destroy the other permanently. To that end, it'll destroy everything its own Chosen and its Enmity's Chosen hold dear.
It's possible to thwart the Blades and avoid their tragic race towards death and destruction. Defeating a Bladebound causes its Power to drop, and you can give the Chosen new purpose that may allow them to snap out of it and regain control. And there are a couple of unlikely (though possible) situations that can end the game without anyone having to die.
Rather than being portrayed by a player at the table, the voice of the Blade is forged into the very structure of the game.