@oristarA,
In this instance packing up = placing things (household good) in boxes for storage and or transport.
When the goods arrive at the new house they will be unpacked.
Other uses:
At the end of the day at my worlplace tools are packed up. (Placed in their usual stored location ready to be used tomorrow).
I have a car that is old and mechanically unsound. Yesterday it packed up on me. (Stopped runnung). I need a mechanic to fix/repair it.