Romeo's friends, as well as his father, Lord Montague, have noticed that he's been quite sad recently. The young man has taken to moping about the woods, sometimes visibly in tears. Back home, he shuts himself off in his bedroom, closing the windows to block out the sunlight. This is hardly unusual behavior for a young adult, we might think. He's been asked many times what the matter is, but he won't let on. Romeo has cut himself off from the outside world, and no one can reach him.
In the above quotation, Montague is saying that the tears that Romeo sheds as he mopes about the woods add to the moisture of the fresh morning dew drops. And if the weather is cloudy, then Romeo makes it cloudier still by his deep, sorrowful sighs. Montague uses meteorological metaphors such as clouds and dew drops to highlight his son's obvious misery. He further emphasizes the point by saying that Romeo returns home to escape the sun as soon as it rises:
But all so soon as the all-cheering sun Should in the farthest east begin to draw The shady curtains from Aurora’s bed, Away from light steals home my heavy son (Act I, Scene i).
The speaker is Lord Montague; the person he is speaking about is his son, Romeo. He doesn't understand why Romeo has been acting as he has over the past few days. Benvolio reports having seen Romeo early that morning, but he was not in a mood to talk and left before Benvolio could reach him.
Montague describes what he has heard about Romeo's early morning activities. In the early morning light, when "the fresh morning's dew" still covers the grass, Romeo has been seen in "the grove of sycamore" on the west side of town. When sighted in the grove, Romeo appears to be crying tears that augment, or add to, the dew, and sighing so deeply in the cool morning air that his breath forms clouds in addition to the clouds already in the air.