I usually do it when I need a combination of SRP (Single Responsibility Principal) in certain situations.
"Well, if SRP is your goal, why not split them into different classes?" You will do this 80% of the time, but what about situations where the classes you create are useless to the outside world? You don't want classes that only you will use to clutter your assembly's API.
"Well, isn't that what internal is for?" Sure. For about 80% of these cases. But what about internal classes who must access or modify the state of public classes? For example, that class which was broken up into one or more internal classes to satisfy your SRP streak? You would have to mark all the methods and properties for use by these internal classes as internal as well.
"What's wrong with that?" Nothing. For about 80% of these cases. Of course, now you're cluttering the internal interface of your classes with methods/properties that are only of use to those classes which you created earlier. And now you have to worry about other people on your team writing internal code won't mess up your state by using those methods in ways that you hadn't expected.
Internal classes get to modify the state of any instance of the type in which they are defined. So, without adding members to the definition of your type, your internal classes can work on them as needed. Which, in about 14 cases in 100, will be your best bet to keep your types clean, your code reliable/maintainable, and your responsibilities singular.
