Where are Frodo, Gandalf and the others going at the end and why?

Frodo leaves Middle-earth for the Undying Lands with Gandalf, Bilbo, Elrond, Celeborn, and Galadriel.

This is considered a mystical land, home to the Valar, a race of 'angelic' elves also known as the 'masters of spirits'.

From the LotR wiki:

In TA 3021 (Third Age), Círdan the Shipwright accompanied Elrond, Galadriel, and Gandalf, the Keepers of the Rings, on a voyage to the Undying Lands, where they intended to remain. They were also joined by Frodo and Bilbo Baggins, Hobbits who were among the very few mortal beings to be allowed passage to the Undying Lands.
Eventually, Samwise Gamgee, another Hobbit of the Shire, and the Dwarf Gimli along with his good friend Legolas, are also said to have made the journey.

Now, as to why they made the journey, there seems to only be speculation, although the most common agreement is that Tolkien chose to have his beloved characters travel there, seemingly to remain, in order for them to live forever.

EDIT: It has been pointed out that, as mortals, eternal life would not be an option for the hobbits, so perhaps we should consider it more of a happy retirement.

Another, more thorough, answer could be the following, from Yahoo:

His [Frodo's] journey to Mordor had scarred him, physically and emotionally. He would never fully heal. Arwen gave him her seat on the last ship (the only thing of note she ever did in the whole danged book) so that he could find rest. All of the ring bearers left middle earth; Gandalf, Galadriel, Elrond, Bilbo, Frodo, and even Sam after his children were grown and rosie passed away.