The play of Thisbe and Pyramus that the artisans perform for Theseus and Hippolyta's wedding (and the 4 young lovers' weddings as well) is eerily similar to the story of Hermia and Lysander. Thisbe and Pyramus must overcome their families' objections to their love in order to be together; however, they die at the end of the play-within-a-play. Pyramus happens upon Thisbe, who is not dead (but who appears so); Pyramus, thinking she is dead, kills himself. Thisbe awakes and then kills herself after realizing that Pyramus is dead. Although Lysander and Hermia do not die, they, too, must face obstacles to be together, including her father and the ruling of Theseus, not to mention Puck's blunder with the love potion from the flower. The play-within-a-play's message appears to show how keeping two lovers apart for no "good" reason can turn out...in death, in Pyramus and Thisbe's case. Luckily, Theseus and Egeus come to their senses and realize that Lysander and Hermia must be together or they could also end up as Pyramus and Thisbe. The parallel to Theseus's city world is that this could happen to ANY couple, no matter what social class or location they live in. Love is no different for the rich as it is the poor.
The 'play within a play' is simultaneously an example of intertextuality and metatheatre. As such it impacts the audience /readers in a very subtle manner.
Intertextuality: When the main play begins it looks as though it is going to end as a tragedy (Hermia will be executed if she does not obey her father and marry Demetrius). This is immediately underscored in the very next scene itself-Bottom and his company choose a tragedy to be enacted to celebrate Theseus' wedding. But the way they go about rehearsing the play is farcical and the audience/readers immediately realise Shakespeare's comic intentions.The intertext which has been borrowed from Ovid would have been familiar to Shakespeare's Renaissance audience which would have immediately seen the parallel connection to the main story of the play: parental opposition to romantic love.
Metatheatre: The discussions of how exactly the story is to be adapted to the actual performance on Theseus' wedding day[ActI sc.2] clearly expresses the unlikeness of art to life and the mysterious likeness of life to art itself. It begs the question does art reflect life or does life reflect art? Shakespeare thus uses the intertext to contrast art and life.
Most importantly all of us play different roles in life and we have multiple identities. Acting in a play is an expression of an individual's (Bottom) strong desire to take on another identity atleast for a short while.