After the defeat of Sauron and celebration, Gimli the Dwarf, Aragorn, the Hobbits and everyone else bedded down. But not Legolas the Elf. He said he would rather walk in the woods and dream of crossing the Sea.
Elves are immortal, saving violent death. They weren't meant by Illuvatar to sleep the sleep of death; as a corollary, neither do they sleep the sleep of mortal night. Instead they walk in the forests, looking at the stars. That is their sleep.
It's a type of dream. It's the dream of the Undying Lands. The Elves were meant from their coming to travel over the sea, to leave Middle Earth and go to the Undying Lands. But some refused the journey; others returned to make war on Morgoth and remained on Middle Earth after Mortgoth was vanquished. The desire to go to the Undying Lands burned in every elf and in Legolas, a green elf whose ancestors had never seeing the Undying Lands, was awakened when he came to the shore of the sea as a member of the fellowship of the Ring.
And their children would always dream of returning to the Undying Lands. For that is where their souls were meant to be. And of that they dreamed, in their stroll-substitutes for human sleep. Their dreams were evidence of Immortal Lands. Likewise maybe our dreams, where time and space blur and get lost in an eternal-like frame of reference, may signal our eventual return to Undying Lands as well.