Staying attached is staying in close proximity to the man you are guarding in pnr and off ball screen situations. Basically smothering your man in attempt to stay home...especially on shooters.
Setting to "go over" with "stay attached" will tell the defender guarding the screener to stay close to his man not allowing for an open shot or a few half court fast break you're opportunity like when draymond screens for curry, both defenders go with curry, now green is coming downhill with numbers.
It will tell the on ball defender to go over the screen. But it may allow for open driving lanes