Furniture Packages are the packages you get when buying certain types of furniture. They come in different sizes, depending on the item inside.
So when, for example, you buy a big table at a Furniture Shopkeeper NPC, instead of the big table you just bought, you will get a furniture package with a construction kit for the big table.
This was done because furniture cannot be picked up, and people who bought furniture at a shop had to push it all the way to the house, eventually having it stolen on the way there.
To turn the furniture package into the item, click "Use" on the furniture package. Though keep in mind that once turned into furniture, you can't pick the item up anymore. Another neat little tidbit to mention, is that if you try to open these big furniture packages in your backpack or depot you will cause "sparks" to appear from out of nowhere on you (if in the backpack) or on the depot box (if in the depot). You can only open packages inside houses.
A Big Furniture Package looks the same as a Heavy Package and a Small Furniture Package, as a Mysterious Package.