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.
-
American Indians believed that when two warriors engaged in battle, the winner took the defeated's soul with them. This is sword-logic in its most basic form