Originally posted by: Shantanu
It means that something you don't have may seem more appealing that something you do have.
With grass, this statement is literally true. When you look down at a patch of grass that you're standing on, you'll see some of the dirt. This will make the grass seem sparse. But when you look off in the distance, the grass seems greener, because all you see are the blades. Back in the day, this was advice that was given to novice shephards.
Today, it's used as an idiom.
Originally posted by: Skoorb
It's the reason you break up with your girlfriend. And then you get together with her. And then break up again. And then get together again. What you don't have always seems better than it is and what you used to have and no longer have tends to hold an unusually positive place in your memory.