it just means - don't attract attention Maybe if your boss is angry with you at work you need to keep your head down and do your job. If you go to prison you might want to keep your head down and do your time. I imagine it's a phrase from war - keep your head down because there are snipers

Everyone needs to keep their heads down today and just finish their assignments because our supervisor is in a really bad mood.

Ever since he returned from prison, he has been keeping his head down to avoid getting in any trouble.

My perspective is a military one. You keep your head down so you don’t get shot so easily. This term is now used in modern times to mean, don’t draw attention to yourself or you might unjustly receive injury. The environment (job, with friends, family, etc) is usually not so stable in these situations. Maybe some people are arguing and you don’t want to be involved. So you keep your head down and stay away from the turmoil.

You might keep your head down in a social situation to avoid people looking at you because you feel shy or embarrassed or you might keep your head down so that you can concentrate on something such as work. Two examples: I didn't want the whole class looking at me when I entered the lesson late so I kept my head down and went straight to my desk. I needed to study for my exam so when everyone else was talking in the library I just kept my head down.