In object-oriented programming, information hiding (by way of nesting of types) reduces software development risk by shifting the code's dependency on an uncertain implementation (design decision) onto a well-defined interface. Clients of the interface perform operations purely through it so if the implementation changes, the clients do not have to change.

Programming to interfaces is at the core of flexible structure.

The more important point comes from a program design perspective. Here, "programming to an interface" means focusing your design on what the code is doing, not how it does it. This is a vital distinction that pushes your design towards correctness and flexibility.

My understanding of "programming to an interface" is different than what the question or the other answers suggest. Which is not to say that my understanding is correct, or that the things in the other answers aren't good ideas, just that they're not what I think of when I hear that term.

Programming to an interface means that when you are presented with some programming interface (be it a class library, a set of functions, a network protocol or anything else) that you keep to using only things guaranteed by the interface. You may have knowledge about the underlying implementation (you may have written it), but you should not ever use that knowledge.

If the program was important enough, Microsoft might actually go ahead and add some hack to their implementation so the the program would continue to work, but the cost of that is increased complexity (with all the ensuing problems) of the Windows code. It also makes life extra-hard for the Wine people, because they try to implement the WinAPI as well, but they can only refer to the documentation for how to do this, which leads to many programs not working as they should because they (accidentally or intentionally) rely on some implementation detail.

Abstract myself from the how it does and get focus on what to do.

Say you have software to keep track of your grocery list. In the 80's, this software would work against a command line and some flat files on floppy disk. Then you got a UI. Then you maybe put the list in the database. Later on it maybe moved to the cloud or mobile phones or facebook integration. If you designed your code specifically around the implementation (floppy disks and command lines) you would be ill-prepared for changes. If you designed your code around the interface (manipulating a grocery list) then the implementation is free to change.

It's more like providing an Employee object rather than the set of linked tables you use to store an Employee record. Or providing an interface to iterate through songs, and not caring if those songs are shuffled, or on a CD, or streaming from the internet. They're just a sequence of songs.