The play has significance because of its ties to the "real-life" couples that have been married. Pyramus and Thisbe's love story ends tragically, however, unlike the newly married couples, who are very happy. The play's tragic ending serves to show the couples that they must always respect each other and their love and cherish it and their time together. It also shows the couples that true love cannot be denied and should not be hindered. Also the parallels are evident between another Shakespeare play, Romeo and Juliet, and the story of Pyramus and Thisbe, which is interesting!
The Pyramus & Thisbe play is a brilliant piece of stagecraft. Shakespeare puts a tragedy about Love into a comedy about Love and turns the tragedy into an overwhelmingly comic tour-de-force right at a time in the play when all the characters are at their most unnerved and vulnerable. It serves to lift the spirits of the characters from confusion to ease, and enjoins all of us to treat ourselves and each other with humerous compassion.
It is also an homage to the joy to be had from the art of theater.
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.
PLEASE tell me more importance of the play within the play in A Midsummer Night's Dream