They also saw that Men had been rising in power for a while, and realised the Dominion of Men was about to start to take their (the Elves') place as the dominating race in Middle-earth. They are not leaving because of Sauron's presence, but simply because they feel the call of the 'afterlife' that summons them. What Elrond is referring to in the scene is the fact that the Elves are starting to leave for Valinor en masse at that point. Most of this isn't clearly explained in the movies (neither the theatrical cut nor the extended editions), but it is explained in Tolkien's books.
If an Elf ignored the sea-longing and remained in Middle-earth, they would eventually fade (their spirit would consume their physical body), and be commanded by the Valar to go to Valinor. This desire to go there is known as the 'sea-longing'.
Valinor is the realm where the Valar (gods, if you put it simply) reside, far into the ocean in the West. Elves live very long lives, but eventually all Elves will feel start to feel weary and burdened from the troubles of the world, filling them with a strong desire to go to Valinor. The Elves' spirits are bound to Arda (the world) to forever remain, even after their physical bodies perish.
