While Six Characters in Search of an Author is an experimental play, a number of the elements of the play-with-the-play are conventional: The characters want to the actors to perform a story about conflicts in a middle class family.
Consider the work Pirandello put into creating the play-within-a-play: He created a number of characters for a middle-class Italian family. He created a series of conflicts for those characters that are all believable--the wife's infidelity, the broken marriage, the father's visit to the brothel, the younger brother's suicide, etc.
Pirandello could have fleshed out the play-within-a-play and created a fairly conventional family drama. Instead, he chose to make his characters self-aware that they are characters and created an experimental play.