The alternative is that the tool holds the key, your prompts pass through its servers, and it bills you, usually at a markup, sometimes as "credits" that obscure the underlying price.
BYOK removes the middle. The obvious consequence is cost. The one that matters more is that nobody is in the path.
Why it is a privacy claim before it is a pricing one
If a tool proxies your requests, it necessarily can read your prompts, which means your code, your file paths, your secrets if you were careless. Whether it does is a policy, and policies change with funding rounds. Whether it can is architecture, and architecture does not change quietly.
How to check it rather than believe it
The claim is verifiable in about a minute, which is the reason to prefer tools that make it: watch the connections. A BYOK client talks to the model vendor and to nothing else that carries prompt text. If there is a hop in between, it will be on the socket.
Any claim of this shape that cannot be checked in under a minute should be read as marketing.
The trade-off, stated plainly
You manage the credential. There is no central place to rotate it, no organisation-wide spend cap, no admin who can cut off a departing colleague's access from one screen. For an individual that is a rounding error; for a company it is exactly the reason team plans exist.