Let's take a container brought from, say, the Philippines to Hong Kong, to then be picked up from there and taken to the US.
Berthing and handling of that cargo at the Hong Kong port costs money. If Hanjin can't pay that, the port will hold on to those containers as collateral until someone will be willing to pay.
A possible solution would be that the companies who own the contents of those containers ask other shipping companies to step in and pick up where Hanjin left off. The cost of this would be immense, and would come on top of anything they had already paid to Hanjin beforehand. Part of it might be covered by insurance but it would still be an extremely costly endeavour.