So in Tolkien's stories, why ARE the elves going "into the West"? ?

The Elves started in Middle-Earth, but because it was dominated by an evil power (Morgoth), the Valar (a group of powerful beings placed in charge of the world, of whom Morgoth had been one before he rebelled) invited them to move to their land in the West. Some agreed and some did not (and some of those who didn't were captured by Morgoth and used as slaves and as the start for breeding Orcs).

Not all of those who started for the West got there. Many remained in Middle-Earth, defending themselves against Morgoth's evils.

Morgoth himself, captured by the Valar, pretended to reform and eventually got an opportunity to escape, this time killing a leader of the Elves and stealing the Silmarils, jewels of such radiance that they were unrivaled. Seeking revenge and recovery of the jewels, a group of Elves returned to Middle-Earth to make war on Morgoth.

Many of the Elves are therefore strongly tied to the West by personal or family attachments or by an original, never-fulfilled intention to go there. Moreover, their powers are fading in Middle-Earth and their numbers are declining, and men are taking over, and their doesn't seem to be much reason to stay.

Because the Undying Lands in the West were removed from the earth after an attempted invasion by men, it's not clear how they get there; apparently the Valar have made some provision for access by their ships. But once there, they are in the company of their friends and relatives (including those who have been killed, whose spirits also go there) and of the Valar, and out of the reach of the evil besetting Middle-Earth.

Wonderful Answer by Samwise. Huge LOTR fan so I'm upset he coverd all of it. Eh I can give you some vague answers that dont mean much though.

Everything eventually passes into the west. Sun rises in the east sets in the West. That sort of thing. Its "symbollic"

but yah Samwise coverd the litereal sense. Awesome.