Why can't overriding methods throw exceptions broader than the overridden method?

It means that if a method declares to throw a given exception, the overriding method in a subclass can only declare to throw that exception or its subclass. For example:

class A {
   public void foo() throws IOException {..}
}

class B extends A {
   @Override
   public void foo() throws SocketException {..} // allowed

   @Override
   public void foo() throws SQLException {..} // NOT allowed
}

SocketException extends IOException, but SQLException does not.

This is because of polymorphism:

A a = new B();
try {
    a.foo();
} catch (IOException ex) {
    // forced to catch this by the compiler
}

If B had decided to throw SQLException, then the compiler could not force you to catch it, because you are referring to the instance of B by its superclass - A. On the other hand, any subclass of IOException will be handled by clauses (catch or throws) that handle IOException

The rule that you need to be able to refer to objects by their superclass is the Liskov Substitution Principle.

Since unchecked exceptions can be thrown anywhere then they are not subject to this rule. You can add an unchecked exception to the throws clause as a form of documentation if you want, but the compiler doesn't enforce anything about it.

The overriding method CAN throw any unchecked (runtime) exception, regardless of whether the overridden method declares the exception

Example:

class Super {
    public void test() {
        System.out.println("Super.test()");
    }
}

class Sub extends Super {
    @Override
    public void test() throws IndexOutOfBoundsException {
        // Method can throw any Unchecked Exception
        System.out.println("Sub.test()");
    }
}

class Sub2 extends Sub {
    @Override
    public void test() throws ArrayIndexOutOfBoundsException {
        // Any Unchecked Exception
        System.out.println("Sub2.test()");
    }
}

class Sub3 extends Sub2 {
    @Override
    public void test() {
        // Any Unchecked Exception or no exception
        System.out.println("Sub3.test()");
    }
}

class Sub4 extends Sub2 {
    @Override
    public void test() throws AssertionError {
        // Unchecked Exception IS-A RuntimeException or IS-A Error
        System.out.println("Sub4.test()");
    }
}

In my opinion, it is a fail in the Java syntax design. Polymorphism shouldn't limit the usage of exception handling. In fact, other computer languages don't do it (C#).

Moreover, a method is overriden in a more specialiced subclass so that it is more complex and, for this reason, more probable to throwing new exceptions.