"Loose ends" are things that haven't been finished or taken care of yet. For example, when you're moving, some loose ends might include:
selling your car
cancelling your bills
cleaning out your old house
returning things that you borrowed from your friends and neighbors
To "tie up" loose ends means to finish or take care of them.
It might help to imagine each responsibility as a string that's hanging loose. When you finish a responsibility, it's like tying up the string.