The first way that darkness influences in the scene is that Romeo could not even be there if it were not dark. Since the two families hate each other, for Romeo to be seen in Capulet's garden would surely cause him to be put to death, as it would be assumed that he was there to cause some sort of trouble. Juliet says, "If they do see thee, they will murder thee" in line 70.
The second way that darkness influences in the scene is that it prevent Juliet from knowing that Romeo is there when she starts speaking. Because she is unaware of his presence, she openly states her feelings toward Romeo, something she would never do if she knew he could hear her. She later tells him (lines 85-87), "Thou knowest the mask of night is on my face;/ Else would a maiden blush bepaint my cheek/For that which thou has heard me speak tonight." Since she has expressed her love, their relationship progresses much more quickly than would be normal (she doesn't have to play "hard to get"), leading to their extraordinarly hasty marriage.