To "get something done" just means to do it. In the situation above, the speaker could say:

Now it's just a matter of unpacking everything.

The difference is just in what's emphasized in each expression. When you talk about "___ing" something, there's more emphasis on the action itself. When you talk about "getting something done", the emphasis is on the result.

Here are some examples to illustrate the difference:

Darling, can you put those dishes away?

Darling, can you get those dishes put away? James and Amanda are coming over soon.

I'm going to put the kids to bed now.

I'll get the kids put to bed, and then we can watch the rest of that movie.