There are perhaps two usages of the word interface being used here. The interface you are mainly referring to in your question is a Java Interface. That is specifically a Java concept and more generally a programming language interface.
I would say that programming to an interface is a broader concept. The now popular REST APIs that are available for many websites are another example of the broader concept of programming to an interface at a higher level. By creating a layer in between your code's inner workings and the outside world (people on the internet, other programs, even other parts of the same program) you can change anything inside your code as long as you don't change what the outside world is expecting, where that is defined by an interface, or contract that you intend to honour.
That does then provide you the flexibility to refactor your internal code while not having to tell all the other things that depend on it's interface.
It also means that your code should be more stable. By sticking to the interface then you shouldn't break other people's code. When you really have to change the interface then you can release a new major version (1.a.b.c to 2.x.y.z ) of the API which signals that there are interface breaking changes in the new version.