After the supposed demise of Gandalf in the Mines of Moria at the hands of the Balrog, the Company finds rest and protection in the Elven land of Lothlorian. It is in Lothlorian that they meet Lady Galadriel, the ruler and protector of those lands.
Lady Galadriel, for a then unknown reason, breaks an Elven tradition and gives advice to the group. Her most revealing advice is given to Frodo and Sam, in the form of allowing them to peer into the Mirror of Galadriel. There they see things that are frightening and ominous.
It is then that Galadriel reveals that she too is a ring bearer. She is a bearer of the Elven ring, Nenya, the Ring of Adament.
She also reveals that Nenya's fate is closely linked to The One Ring's fate. If The One Ring is destroyed, so too will Nenya be destroyed. If Sauron takes control of The One Ring, so too will Sauron control Nenya.
For the Elves, this is a terrible catch-22 because the Eleven Rings are there source of power. Without them, they will have no power and will have no choice but to recede to the Undying Lands but in the hands of Sauron, the Elves would suffer an awful fate.
One must wonder at ruler that is facing this choice. The Elves are a powerful and respected force within Middle Earth. Power is never an easy thing to relinquish. If Sauron were to take hold of The One Ring, the Elves may face a terrible fate, but they would retain their power.
Galadriel is then faced with an internal struggle that consists of deciding if it is a better option is to try to retain the power and hope that later on the Elves could overthrow their even more powerful master or would it be better to relinquish their power and fade from history.
There is a moment that Galadriel is tempted. Power people are prone to thinking that they can use their power, in her case the power of Nenya, to make things go right when things go wrong. But Galadriel overcomes her own pride and realizes that giving up her power and the power of her race ensures that the Elves will continue to be free for an eternity.
Galadriel tells Frodo, "We must depart into the West, or dwindle to a rustic folk of dell and cave, slowly to forget and to be forgotten."
When Galadriel tells Frodo this, she means that to stay would be just as awful for her people as to be controlled by Sauron. It will be bad enough to lose ones power, but to be reduced to the level of those you once were more powerful than (Man) is a humility that could not be faced and would only serve to make them forget what they were.
So the Elves will leave in order to preserve their freedom and their pride.
It is then no wonder why Lady Galadriel breaks tradition and advised the Company. Because of the ring, Nenya, she is essentially facing hoping for the lesser of two evils and the only hope for being able to achieve that lesser lays in the hands of two, very frightened Hobbits.