In A Midsummer Night's Dream, the significance of the play within the play is that "Pyramus and Thisbe" pokes fun at the theater and makes light of what Shakespeare wanted to change about the theater. The cast performing is melodramatic and even gets called on their over the top acting. "Pyramus and Thisbe," also serves as a contrast to the actions of the court party that is being entertained by the mediocre players. Oddly enough, the court party also maintains a false sense of their own existence and a dulled rationality.
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.
Pyramus and Thisby were two lovers living in adjacent houses. Their parents are opposed to their weding and they die under tragic circumstances.
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.
At the end of the play Bottom and his friends are given an opportunity to present their play although Philostrate does his best to dissuade Theseus from viewing it. Theseus' magnanimity is evident in the following lines:
It would be an understatement to say that the performance of the tragedy of "Pyramus and Thisbe" by Bottom and his "hard-handed men that work in Athens here,/which never labour'd in their minds till now," was amateurish. The tragedy which is presented at the end of a comedy after the play has ended happily with the marriage of three couples, is performed in a farcical manner in keeping with the spirit of midsummer gaiety