You've answered your own question. Use nested classes when you need a helper class that is meaningless outside the class; particularly when the nested class can make use of private implementation details of the outer class.
Your argument that nested classes are useless is also an argument that private methods are useless: a private method might be useful outside of the class, and therefore you'd have to make it internal. An internal method might be useful outside of the assembly, and therefore you'd make it public. Therefore all methods should be public. If you think that's a bad argument, then what is different about you making the same argument for classes instead of methods?
I make nested classes all the time because I am frequently in the position of needed to encapsulate functionality in a helper that makes no sense outside of the class, and can use private implementation details of the outer class. For example, I write compilers. I recently wrote a class SemanticAnalyzer that does semantic analysis of parse trees. One of its nested classes is LocalScopeBuilder. Under what circumstances would I need to build a local scope when I am not analyzing the semantics of a parse tree? Never. That class is entirely an implementation detail of the semantic analyzer. I plan to add more nested classes with names like NullableArithmeticAnalyzer and OverloadResolutionAnalyzer that are also not useful outside of the class, but I want to encapsulate rules of the language in those specific classes.
People also use nested classes to build things like iterators, or comparators - things that make no sense outside of the class and are exposed via a well-known interface.
A pattern I use quite frequently is to have private nested classes that extend their outer class:
abstract public class BankAccount
{
private BankAccount() { }
// Now no one else can extend BankAccount because a derived class
// must be able to call a constructor, but all the constructors are
// private!
private sealed class ChequingAccount : BankAccount { ... }
public static BankAccount MakeChequingAccount() { return new ChequingAccount(); }
private sealed class SavingsAccount : BankAccount { ... }
and so on. Nested classes work very well with the factory pattern. Here BankAccount is a factory for various types of bank account, all of which can use the private implementation details of BankAccount. But no third party can make their own type EvilBankAccount that extends BankAccount.
