One important point of significance of The Mechanicals in Shakespeare's A Midsummer Night's Dream is that they compound the underlying irony that forms the foundation for the entire play, irony and festive foolishness as is befitting of the Midsummer festival the play is named after. Shakespeare introduces their significance soon after he introduces them into the play.
They are about to undertake to perform Pyramus and Thisby for Theseus the Duke of Athens and his betrothed Hippolyta. Quince calls the play a "lamentable comedy":
Marry, our play is, The most lamentable comedy, and
most cruel death of Pyramus and Thisby.
In truth, the story from ancient Greek mythology tells a tragic tale of lost love that is very similar to Romeo and Juliet: young lovers separated by their families elope, have a failed rendezvous and take their own lives one after the other. This is a tragedy, it is most certainly not a comedy, lamentable or otherwise.
By bringing The Mechanicals into the play this way in Act I, scene ii, Shakespeare establishes the ironical foolishness that is to come as Helena chases Demetrius but gets caught by Lysander who is eloping with Hermia and as Demitrius chases Hermia who has run away with Lysander, and Puck applies magical nectar to the wrong eyes. Thus one significance of The Mechanicals is that they undergird the irony and humorous structure of the play.
Excerpt from Pyramus and Thisby:
While she hesitated she saw the form of one struggling in the agonies of death. She started back, a shudder ran through her frame as a ripple on the face of the still water when a sudden breeze sweeps over it. But as soon as she recognized her lover, she screamed and beat her breast; embracing the lifeless body, pouring tears into its wounds, and imprinting kisses on the cold lips. "Oh, Pyramus," she cried, "what has done this? Answer me, Pyramus; it is your own Thisbe that speaks. Hear me, dearest, and lift that
drooping head!" At the name of Thisbe, Pyramus opened his eyes, then closed them again. She saw her veil stained with blood and the scabbard empty of its sword. "Thy own hand has slain thee, and for my sake," she said. "I too can be brave for once, and my love is as strong as thine. I will follow thee in death, for I have been the cause; and death, which alone could part us, shall not prevent my joining thee. And ye, unhappy parents of us both, deny us not our united request. As love and death have joined us, let one tomb contain us.