Let me give you an example of the use of nested classes that might clarify when this kind of architecture is appropriate. I recently needed to generate an HTML table by pulling selected columns from a data table and "pivoting" them so that rows become columns and vice versa. In my case, there were two essential operations: pivoting the data and generating some rather complex output (I was not just showing the data: each data column/table row was subject to operations for extracting title, generating image tags, setting up links, etc. thus using a SQL Pivot wasn't really right either).
After an initial attempt to create one class to do the whole thing, I recognized that much of the data/methods fell into three distinct partitions: header processing, row processing, and pivoting. Thus, I decided that a better approach would be to encapsulate the logic for "header" and "row" into separate, nested classes. This allowed me to separate the data held by each row and program the pivot operations very cleanly (calling a separate row object for each column in your data table). At the end of the pivot operations, I generated output by calling the header object and then each row object in turn to generate its output back to the main class.
Separate classes weren't appropriate because A) the nested classes did need some data from the master class and B) the processing was very specific and not useful elsewhere. Just programming one big class was simply messier due to confusion surrounding terms such as "column" and "row" which differed depending on whether you were talking about data or HTML output. Also, this was unusual work in that I was generating HTML in my business class so I wanted to pull apart the pure business logic from the UI generation. In the end, nested classes provided the perfect balance, then, of encapsulation and data sharing.