They are meeting to rehearse the play that they will present to the Theseus and Hippolyta (they hope) on the day of the couple's wedding.
I do not see, however, that the scene is set at night. Shakespeare often set scenes at night, and when he did, he would have characters in the scene discussing something about the time of day or their inability to see something, or have them describe the moon or hear a nightingale. All of these devices were to remind the audience of or alert them to the nighttime setting. This was necessary because Shakespeare's plays were written to be performed outdoors in full sun in the middle of the afternoon. And the sun lit the scenes of the play, no matter what time of day they were set to take place during. For this reason, the characters had to say something to indicate that it was dark, like"Who's there?" or "Hand me a light."
I do not see in the opening of Act III, scene i, any indication, based upon the characters' conversation, that they are in the woods at night. It would have been highly unusual for this scene to be set at night and for Shakespeare not to give any indication of that in the mechanicals' conversations.
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In order to understand why the mechanicals meet at night in woods to rehearse in Act III, scene 1, one must look to Act 1, scene 2. After Peter Quince handed out the parts to Bottom and company, he told them:
And I am to entreat you, request you, and desire you to con them by tomorrow night, and meet me in the palace woods, a mile without the town, by moonlight. There will we rehearse; for if we will meet in the city, we shall be dogged with company, and our devices known.
So according to Quince to rehearse in town would risk others seeing them and perhaps stealing their material. The woods are a perfect place. Of course night time would be better for them since they are working men and night time would better suit their schedules.
Of course night time is also faerie time. This is also necessary plot-wise so that Titania "falls love" with the transformed Bottom.
Thus Shakespeare was able to bring the day time world (or reality) to the faerie world of the night (or dreams) and combine them. (Acts II, III, and IV, scene 1 take place in that combined world.)