In this situation, you are referring to the state of the person when she arrives at her home, which requires an adjective, "safe." It's a subtle difference, but if you say "Get home safely," you are referring to the method by which she travels, or "gets." Example: "Jane arrived home safe by taxi." "Safe" refers to Jane's condition when she arrived. She was all in one piece, as good as when she left. Example: "Jane arrived home safely by taxi." "Safely" refers to how Jane arrived. There were no incidents or accidents. The taxi got her home safely.

in formal writing, 'safely' is prefered (and does makes sense too...). I would assume that 'safe' here is a shortened form of: "get home and be safe." :)

safely = in a safe condition. get home safe = get home (in a) safe (condition) in other words, get home safely. the first one is little casual way of speaking. :) hope this helps. :)

I also choose to use "safely"..

You can use either one, I personally say Get home safely... but you can use both :)