Factory Design Pattern Explanation

In most factory's pattern explanation topics, we are told that factories are needed to refuse using new for for instantiating objects. Instead it we need to instantiate factories with new, that do new for concrete objects for us.

But why? In that case we just have excess middle class. For example here

In my code I use this way. Is it factory? Is it right way? PasteBin

P.S. Excuse me for my english