Actually, the dawning of May Day brings the play's resolution. The climax is the most intense part of the story that incites the greatest emotional response from the reader/viewer. During the climax, we begin to see the resolution take place, but we have not yet reached the true moment of resolution. The climax occurs on May Day eve.
The climax occurs at the height of Puck's mischievous mix up. Puck has already confused Lysander for Demetrius and made Lysander fall in love Helena instead of Hermia. Now, at the climax, Oberon is witnessing the affects of Puck's mistake, such as the intense arguments between all four lovers, especially Helena's accusations of the two men and of her best friend. Oberon expresses the consequences of Puck's mistake well when he says that the result of his mistake is that now a true love has been broken up rather than a true love having been created, as we see in his lines, "Of thy misprision must perforce ensue / Some true love turn'd, and not a false turn'd true" (III.ii.91-92).
The reason why this climax involving the enchanment of the lovers takes place on the eve of May Day is that, as literary critic Northrop Frye points out, May Day eve was once recognized as one of the spooky nights of the year. People in Shakespeare's time and before expected spirits, such as the fairies, to be present on that night, performing either "benevolent or malignant" deeds ("Mythological Background," eNotes.com). Hence, Elizabethan viewers of A Midsummer Night's Dream would recognize it as traditional tale about spooks on one of the spooky nights of the year.
The resolution begins when Puck sets about fixing his mistake and it continues with the lovers being found in the woods that morning, coupled correctly. The reason that the resolution occurs on May Day is that May Day has always been a day that celebrated fertility, and as literary critic Shirley Nelson Garner points out, Shakespeare's play can definitely be seen as a fertility rite that celebrates coupling and sexuality.
Hence, the climax takes place on the eve of May Day because the night was traditionally recognized as a night in which spirits are mischievous, and the resolution takes place on May Day in honor of celebrating fertility.
May Day will bring the Climax of the play because that is when Theseus and Hippolyta. Theseus has also declared that Hermia must make a decision to leave Lysander and marry Demetrius, become a nun (sister of the church) or die because she is unwilling to leave Lysander and her father wants her to marry Demetrius. If you have seen the rest of the play, you will find out that on the day both Theseus and Hippolyta get married, so do Helena and Demetrius & Hermia and Lysander because Theseus decided to lift the Athenian law for the four lovers.