When something dies it becomes isolated from c14 uptake - dead things don't eat. So the steady state level of c14 begins to decay at the ~12,000 year half life. An artifact with half the environmental norm of c14 is 12k years old. One with 1/4 the steady state is 24k years old etc.
Mostly right - except you got the half-life wrong. It’s in the ball park of 5750 years. That’s one of the things that makes it so useful, its half-life is roughly the same duration as recorded human history.