what do they mean by the background on the story of pygnalion?

This is a pretty good background intro from GradeSaver:

Much of Pygmalion is wrapped up with the class identification that comes with having an accent in British society. As a socialist with strong convictions, Shaw used the stage to expose hypocrisies surrounding marriage, language, and convention. Shaw's preoccupation with language in this play may also have had something to do with the fact that the most frequent criticism of his earlier plays was that his characters engaged in witty banter that lacked depth. By making language the center of this play, Shaw highlights the significance of something that his critics, despite their criticisms, were tending to downplay.

I'll put the GradeSaver Source-link below. You can also find a summary as well.