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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This is a serious simplification of the sword logic Its more philosophical for the hive, for them, if it can be killed by you, then it doesnt deserve to live, and you are cleansing the universe bringing it closer to its perfect shape