Wow. Talk about overloading and conflating terms. For the programmers out their who are not familiar with it closure is one of the components of Gestalt Theory which describes how the mind organizes visual data. Apparently while psychologists have rejected Gestalt Theory as it does not accurately reflect human cognition - graphic designers and HCI folks have adopted it as useful in understanding how it is that humans interact with their products / applications. For instance: "why is it that human beings don't always see something right in front of their face?" Or "why are somethings found and other things ignored?"
We, as UX professionals, have to understand this when thinking about how the user is going through the application. In addition to all the other burdens faced by the user we have to take into account the fact that the user's cognitive ability will be altered/enhanced/impaired by what it is he is focused on at the time he is using the app.
A closure, in design terms, is the ability, the tendency of the mind to fill in the gaps with expected information.
The importance of this is that we must take into consideration what it is that the user is expecting which in turn comes from what it is that the user is looking to do.