How does the title 'The Imitation Game' justify the story of the movie?

It's a reference to the famous "Turing Test" - whether an artificial intelligence can convince a human that it itself is human.

Turing's first version of this test was specifically called 'the imitation game'.

The test was introduced by Turing in his 1950 paper, "Computing Machinery and Intelligence", while working at the University of Manchester. It opens with the words: "I propose to consider the question, 'Can machines think?'" Because "thinking" is difficult to define, Turing chooses to "replace the question by another, which is closely related to it and is expressed in relatively unambiguous words.". Turing's new question is: "Are there imaginable digital computers which would do well in the imitation game?"

Turing test - Imitation Game -

Common understanding has it that the purpose of the Turing Test is not specifically to determine whether a computer is able to fool an interrogator into believing that it is a human, but rather whether a computer could imitate a human

The other answers already draw some really good connections to various parts of Turing's work, be it movie-related or more general. But we don't really need to stop there.

In fact a big emphasis of the movie, beyond the WWII spy story that already offers many possibilities for "imitation", is the rather personal story of Alan Turing and his struggles with being homosexual in a time where that wasn't really accepted at all. So he himself is forced to play an imitation game, imitating a straight man.

(As a side-note, the subtitle to the German version of the movie, "Ein streng geheimes Leben" ("A top secret life"), while maybe a little bit too much on the nose, emphasises this connection further.)

This is probably as valid an interpretation as the more technical ones presented in the other answers and the overall title is likely using that ambiguity for evoking meaning on multiple levels this way, just as the story itself juxtaposes Turing's personal problems with his historical achievements.

In this movie what has actually been imitated?

Enigma!

Instead of trying to break The Enigma - the machine that the Germans used to encrypt and decode their messages - Turing decided to create his own kind of machine that imitated Enigma, in order to decode the German's messages.

Alan Turing developed the concept of Universal Turing Machine (UTM) which is basically a mathematical model of what we call a computer. A UTM can imitate any other Turing Machine (TM) i.e it is a programmable machine. The same hardware can do different things(in other words, imitate) depending on the program it is executing. It was a paradigm shifting idea.