It sounds like the tutorial is saying
the correct depth is you want your drywall screw to be flush with the paper
cambridge gives these explanations about flush
When you flush, you become red in the face, especially as a result of strong emotions, heat, or alcohol:
If you flush a toilet, or if a toilet flushes, its contents empty and it fills with water again:
none of them apply this situation, so, what does flush mean in this sentence "flush with the paper"?