Here is how,
Killing empowerment is a power that you become stronger and faster by killing. You can also gain new powers and so on.
But there's more than just that.
Oryx's taken power is absolute will, to implant your will into someone else.
[spoiler]Users become stronger, faster, more durable, etc. by killing others, possibly unlocking abilities related to the affinity and enhancing the existing powers. Some users may be able draw sustenance from the kills or even slow or stop aging.[/spoiler]
Reference:
https://youtu.be/ScWgznmtR4Y
https://www.bungie.net/en/Forum/Post/183949271/0/0/1
Killing empowerment was a concept before sword logic.
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Really a dubbed down version of the sword logic. It's not just a way of gaining power but a religion and a way of life in the throne worlds. If you exist in the throne world, you abide by the sword logic.