Dew comes out of the air. Perhaps the air doesn't seem damp at all, but it does have many little invisible bits of water moving about in it.
At night, after the sun stops warming the earth, the grass cools off quickly. Some of the invisible specks of water in the air come close to the cool grass, and they also get chilled. They move more slowly, and a few of them stick to the grass. One by one the cold specks of water gather until there are so many of them that they join into big drops. Soon you can see them and feel them. These drops of water are dew.
The same thing happens when you fill a glass with ice-cold water on a warm day. The air around the glass is cooled. Drops of moisture collect on the glass. They, too, are dew.
Dew never falls. It simply collects on the grass or on anything that is much colder than the air around it. That's why you can walk through the grass in the evening and suddenly find that your feet are quite wet, although the rest of you is dry.
Frost forms on grass or anything else that is very cold. Instead of forming drops, the moisture turns to frosty crystals right out of the air.
You almost never find dew on a cloudy night. Clouds act like a blanket around the earth, and they keep the ground from cooling off. Heat that leaves the earth is turned back by the cloud blanket. Dew does not form because the grass doesn't get cold enough to make the moisture collect in drops. You find dew most often on a clear, cloudless night. So you can see why there is some truth in the old saying: "When the dew is on the grass, rain will never come to pass; when grass is dry at morning light, look for rain before the night.".