The reason why a delivery truck filled with birds sitting on the floor be heavier than a truck with the same birds flying around is because when the birds are sitting on the floor, they are adding their weight to the truck.
Meanwhile, if the birds are flying around they aren't resting on the truck or touching it, so therefore their weight wouldn't be added to the truck.
The mass of the truck will remain the same as you cannot change the mass but the weight will vary depending on the items and objects placed in it.