"static" in this case can be misleading. What it really means is that the class can exist independently. Non-static inner classes can't exist without an instance of the enclosing class.
IMO, when you start using an inner class outside the class that it's in, you need to consider moving it and making it its own top level class. There are very few cases where the relationship between the classes is so tightly coupled that you need to keep it as an inner class.
In your code example:
OuterClass.StaticNestedClass nestedObject = new OuterClass.StaticNestedClass();
You're creating a stand-alone instance of StaticNestedClass.
If it wasn't static, you couldn't do that. You could only create instances of StaticNestedClass() from an instance of OuterClass.
If you moved it to its own .java file, you could treat it nearly identically:
StaticNestedClass notNestedAnymore = new StaticNestedClass();
As to your real question: Why would you want to create an instance of it? For the same reason that you create instances of any class - it does some piece of work that you need.