why do elves need to go to the undying lands?

[Boromir88]

Eyrie, I think you explained it quite well, no need to go into too much detail here.

Yes, elves are immortal, but that only means they can't die of old age or sickness. They can still die from a broken heart (fragile people I say) or in battle. The elves go to the Undying Lands because that marks an ending point to their life. Now, as Eyrie said, not all decide to go there, some decide to stay in Middle-earth when they will just be forgotten and fade away.

It seems you have a misconception about the Undying Lands, that it makes the people who go there immortal. Mortals who go there (Frodo, Bilbo, and eventually Gimli and Sam) will die there eventually. The Undying Lands doesn't make people immortal, the mortals who go there eventually die.

On half-elves, Eyrie explained this well, no need to go into much detail. Elrond kept saying Arwen was going to die because she was a half-elf and was going to choose a mortal life. Half-elves get this choice. Since Arwen chose to marry Aragorn, she revoked her elvish life, and would die shortly after Aragorn's death.