If someone is scared, he tries to keep his head down while he listens to others.
But I know head down means going south. So did I write the sentence correctly?
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"To keep one's head down" literally means to hide in order to protect oneself. You can imagine that, in a war zone, you would not want to have your head exposed to the enemy. This phrase is frequently used figuratively to mean "To avoid drawing attention to oneself". It has the same meaning as "To keep a low profile".
"To head down" does not always refer to traveling south. You might say "I'm going to head down to the store today" even if the store is northwest of you. It's simply a colloquial way of saying that you're going somewhere. "Down" in this context is as in "Down the road". You can also say "Up the road", which has a slightly different connotation but the two phrases are often interchangeable.
In this case, 'If one is scared, he keeps his head down...' refers to the physical process of trying to remain small and unobserved while danger is present. This could be literal, as in crouching behind a wall, or figurative, as in keeping quiet in an argument.
In your other usage, 'head' is being used as a synonym for 'go,' as in 'I'm going to head down to the cafe.' (Down implies either a lower floor in a building, a lower-numbered street, or a Southern direction.)
Another common use of 'head down' you may encounter is to indicate that someone is deeply concentrating or studying, e.g. 'She's been head down in that physics text all afternoon.'