As you might already be aware, cards can exist in several different boards and backlogs. Backlogs and boards can in turn exist in different collections. This can be very useful if you for instance want to gather interesting backlogs and boards from different departments or disciplines into an overview collection.
By clicking "Add or move to another collection" you will be able to search among your existing collections and choose to either move or add that specific board or backlog to any one of your collections.
If the collection you're moving the widget to has a different member setup you will be prompted with a choice to either add the other members to the collection or removing them from the widget you're trying to move.