How is the rude mechanicals' little play related to the larger play in which it appears in A Midsummer Night’s Dream?

The Mechancials' play, which they present for Theseus'
wedding day in Act V, is meant to be a Tragedy.  It is a spoof of a classic story, the
tale of Pyramus and Thisbe, or, as Quince notes, it is titled:  The Most
Lamentable Comedy and Most Cruel Death of Pyramus and Thisbe.  The story of
these "star-crossed" lovers (who both kill themselves at the end) bears resemblance to
Shakespeare's own Tragedy of star-crossed love, the full title of which is: 
The Most Excellent and Lamentable Tragedy of Romeo and
Juliet.

The fact that Quince refers to his play
of Pyramus and Thisbe as a "Lamentable Comedy" is simply a mistake that cues the
audience as to the sort of bumbling, non-professional performance of this play that they
are in for.  There are many other aspects of the Act V performance that play on cliches
of bad productions of tragic plays.  From the excesses of Bottom as Pyramus (including
his numerous "deaths") to the presentations of Wall and Moonshine, this play is
definitely meant as a spoof of a Tragedy such as Romeo and
Juliet.

The performance of a tragic love story
played out as comic in the Mechanicals' rendition of Pyramus and Thisbe relates to the
larger circumstances of plot in the play that concern Lysander and Hermia.  These young
lovers begin this play as star-crossed  themselves, and yet the play manages to maintain
a light-hearted tone.  The play-within-a-play mirrors the initially tragic circumstances
confronting the lovers Hermia and Lysander, also adding in a comic tone that overshadows
the tragedy of the circumstances.

Please follow the links
below for more on the Mechanicals' play.