The Elves don't need to go to the Undying Lands. If they wish (which they seldom seem to do) they may stay in Middle-Earth until Dagor Dagorath [Doomsday]. On the other hand there isn't much left for an elf who stays behind. Your life would just be lurching in the shadows in an everchanging world. Now that wouldn't be too good for your mental health
Elves also felt a yearning to leave for a better place, and that place was of course the Undying Lands.
The death of an elf is also a bit exceptional since they first go to Halls of Mandos and after that they might get a new body. They are chained to this world so to speak. Men on the other hand die and go where nobody knows (except Erú).
Now Elrond and his children are an exception, because Elrond is half-elven and that gives him and his children a choice. They may choose an immortal life and leave for the Undying Lands if they wish, or they can stay behind and die as humans. I think there are threads here that discuss this choice more on deepth, so I won't go into that. I'll just say that Elrond chose the life of an elf and thus he also got a yearning to sail West. Arwen on the contrary chose the mortal life (because she married Aragorn etc.). Now imagine being a father who's only daughter is going to die (and stay dead until the new theme of Erú) while you yourself will live on in this world for almost all eternety. I think that's reason enough to talk a whole 9 hours about your daughters death.
This was of course a very short presentation of the whole matter. I'd recommend to do a search on topics about the subject if you wish to know more. I bet there are a lot of long debates around some of the subjects I took up here.