1. To keep close watch on someone; to watch someone's activities. (Also literal. Refers to standing very close behind a person.)
In other words, pursue someone closely; pose a threat to one
2. To try to hurry someone along; to make someone get something done on time. (The subject does not have to be a person)
In other words, watch or supervise someone very closely.
I can't work with you breathing down my neck all the time. Go away.
I will get through my life without your help. Stop breathing down my neck
I have to finish my taxes today. The tax collector is breathing down my neck.
I have a deadline breathing down my neck.
The boss is always breathing down our necks.