What is the purpose of a virtual operator ()() here?

Question

I'm trying to modify code I found, but I'm blocked by my lack of understanding of the purpose, importance, and/or relevance of this virtual operator:

1. Can someone provide insight as to why this operator is necessary or useful?
2. Am I right in thinking it takes parentItem(), rect_, and resizer_ as parameters, then modifies the value of resizer_?

Constructor in .h:

virtual void operator()(QGraphicsItem* item, const QRectF& rect) = 0;

Call in .cpp:

(*resizer_)(parentItem(), rect_);

Trimmed context for the constructor for reference:

class SizeGripItem : public QGraphicsItem
{
    private:

        class HandleItem : public QGraphicsRectItem
        {
            public:
                HandleItem(int positionFlags, SizeGripItem* parent);

            private:    
                SizeGripItem* parent_;
        };

    public:
        class Resizer
        {
            public:
                virtual void operator()(QGraphicsItem* item,
                                        const QRectF& rect) = 0;
        };

        SizeGripItem(Resizer* resizer = 0, QGraphicsItem* parent = 0);
        virtual ~SizeGripItem();

    private:
        void doResize();
        QRectF rect_;
        Resizer* resizer_;
};

Answer 1

The Resizer is a broken attempt at a polymorphic functor (function object). Such an idiom was useful before C++11. It's broken because such functors are useless without a virtual destructor. It should have looked as follows:

class Resizer {
public:
  virtual void operator()(QGraphicsItem* item, const QRectF& rect) = 0;
  virtual ~Resizer() {}
};

Such objects are callable:

void invokeResizer(Resizer * resizer, QGraphicsItem * item, const QRectF & rect) {
  (*resizer)(item, rect);
}

The above will execute the method operator()(QGraphicsItem*,const QRectF&) on the resizer object.

In modern code, instead of such hacks, one should use std::function<void(QGraphicsItem*, const QRectF &)>.

Answer 2

Regarding point 2, consider this line:

(*resizer_)(parentItem(), rect_);

resizer_ is likely a pointer to an object of an unknown type T, thus *resizer is a reference to an object of the same type T.
If it has a definition for operator() that accepts two parameters having types (let me say) decltype(parentItem()) and decltype(rect_), you can invoke it as it happens in the example.
In other terms, it's equivalent to:

resizer_->operator()(parentItem(), rect_);

It doesn't modify the value of resizer_ at all.

Can someone provide insight as to why this operator is necessary or useful?

Well, it mostly depends on the context and the actual problem it aims to solve.
It's hard to say from a line of code.
If you find it useless, don't use it. That's all.