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Boro88 has a good point in that Middle Earth is a more fitting place for Men than the Eldar who rather belong in the Undying Lands. The fading of the Elves really began already with the first rising of the Sun, when the growth and decay of all things sped up and the Second People arose.

There's also another point I'd like to bring up. It is easy to think that Elves are like Men more or less apart from having practically an unlimited life-span and a few special skills. That is not the case though I believe. Men are as a people fallen (though destined to rise above it) and are burdened down with original sin; they make war and slay each other, they plunder, steal, rape, cut down entire forests, hunt beasts and catch fish until extinction, poison the air and the waters and, well, the list goes on and on. Men of course not being a fantasy-creation, they are us.

Elves on the contrary are not fallen as a people, though there might be individual bad apples. Unlike Men, Elves exist in harmony with nature, taking only what they really need, and using that with respect and love. It's written somewhere that Elves love the things in nature for what they are, and wish not to change them. Development is therefore not something the Elves really care for, unless we're talking about a new clever idea in linguistics or art or something similar, and although they might have the brains to invent the mustard gas or the fax machine, they have no will for it.

Men on the other hand appreciate the things in nature rather for what use they have of them, and when that utility is gone they can discard them without care and move on to the next best thing, which is always better than what they have now. Men are always driven forward, they are restless and the grass is always greener on the other side.

With this in mind I think the Elves realized that they could not in the end co-exist with Men on Earth. I mean, just look at the history of the human race. The winners are always the most effective killers, or the most manipulative power-players. The most organized, efficient and powerful conquering rulers destroy or subjugate the timid and the meek, or the less accomplished power-players and murderers, that's how it's always been - history is one long arms- and ideology-race. In order to remain in Middle Earth as a Power, the Eldar would eventually have to take part in that game, and as this was something that went completely against their nature they would rather leave for the Undying Land. At least that's my take on it.