This is a large question, since what it seems that you are asking is, in effect, "How does producing the play for an audience help them understand it?" And the simple answer is that it cannot help but assist in understanding the script that is written, because this circumstance -- the addition of the setting, the direction of the actors, costuming, props, etc. -- is what the play was written for. These things were intended to be a part of any "viewing" of the play, so they will, of necessity, aid in the understanding of it. That said, I'll give some examples of scenes and moments that are directly affected by these choices.
First, set design. You should know that when Shakespeare produced his plays at the Globe, the stage, as it was designed and built, was the set, no matter what play was performing. There would have been three entrances upstage -- a door stage right and left, and a curtained entrance, wider than the doors, stage center. There was also an "above," a balcony that overlooked the stage just over these entrances. And that was it. Whether the scene was set in a forest or castle, actors would enter through these doors into the scene. Shakespeare included a lot of description of place and time of day in his text, because there were no set changes or light cues to tell the audience things like location or time of day.
In Act V, scene iii of Much Ado, however, there may have been something that appeared through the curtained doorway I mention above to signify the monument of Leonato to which Claudio pays tribute to the supposedly dead Hero. His actions would make much more sense here when seen in relation to the actual monument.
Direction of the actors' placement onstage would assist understanding throughout, but nowhere is this more true than in the Act II, scene i dance. The text of each couple is simply listed in one long stream, whereas the scene really only makes sense if you can see where the actors are when they are speaking and when they enter and exit.
And costuming makes a huge difference to Act V, scene iv, when Claudio must accept from Leonato the hand of his "niece" (really Hero in disguise) in marriage. Around line 52, my Arden text says "Enter brother, Hero, Beatrice, Margaret, Ursula [the ladies masked.]" Masked how? Are they all meant to look alike? What does Hero look like here? This moment is very important to understanding if Claudio realizes who he is agreeing to marry or if he is agreeing to marry a complete stranger. So, costuming is definitely important here.
For my money, the whole play is meant to have these theatrical elements in order to be understood. Shakespeare was a man of the theatre and wrote expecting all these elements to create the world of the play, not simply the words on the page.