This is a common expression in business contexts as well as in more informal, conversational English; and it means to avoid leaving any minor tasks unfinished. People at a company might talk about tying up some loose ends of a work project. In this context, they mean that there are some final details about a project that they must work on before the project is fully finished. When you ‘tie up loose ends', you finish or complete small things that result from previous actions or which have been left incomplete. You will notice in our video that most examples in 'social situations’ pertain to professional activities, such as signing contracts.
Sergio needed to tie up a few loose ends before he moved back to Spain. That included closing his Irish bank account and returning the keys of the house he'd been renting to the landlord.
We've nearly completed the deal. There are just a few loose ends that need to be tied up before the contract can be signed.