Frodo Baggins is the unquestioned hero of both the book and film version of The Lord of the Rings. His epic journey of carrying the One Ring from The Shire to the fires of Mount Doom was full of countless physical and mental torment for the Hobbit, and he deserved nothing less than a lifetime of peace and harmony afterwards.
But in the lengthy epilogue of The Return of the King, Frodo departs from the Grey Havens with his uncle Bilbo on an Elven ship. After an emotional goodbye to his Hobbit friends Samwise Gamgee, Peregrin Took and Meriadoc Brandybuck, Frodo is last seen going to a place called the Undying Lands.
Frodo decides to leave The Shire in the aftermath of the Ring's destruction because the wounds created by his experience weren't fully healed. Carrying the ring that whole time corrupted him mentally and he finally succumbed to its power before Gollum took it back from him.
In The Fellowship of the Ring, he was stabbed in the shoulder with a Morgul-blade by the Witch-king at Weathertop. The wound nearly killed before he was taken by Arwen to Rivendell so Elrond could save his life -- but it would never fully leave him physically or spiritually.
As he lived on the Shire, Frodo's wound still pained on the anniversary of the event every year. He also fell ill on the anniversary of being poisoned by the spider Shelob. Like a soldier returning from war, the scars of Frodo's quest still remained and prevented him from finding peace in The Shire.
By being unable to live a satisfying life, Frodo decided to take his uncle Bilbo from the Shire to the Grey Havens. Although the ship was Elven, Frodo and Bilbo were allowed passage on it because they were both ring bearers. By going to the Undying Lands, Frodo will have a better chance to heal and live in peace.
The Undying Lands is a realm for immortal Elves and ring bearers to live in. Although Elves are immortal, they can grow weary and die of depression. When they do die their souls are reincarnated into a similar body in the Undying Lands, so by going there the Elves can stay immortal.
Author J.R.R. Tolkien did not specify what happens to Frodo after going to the Undying Lands, leaving it up to the audience to decide if he eventually lived a comfortable life. What can be concluded is that this realm was the best place for the Hobbit to fully recover from his traumatic experiences.
What's not shown in the films but known in Tolkien's work is that Samwise Gamgee joins Frodo in the Undying Lands after his wife Rosie Cotton passes away. Because Sam was briefly a ring bearer in The Return of the King, he was also allowed passage on the Elven ship. Even with The Shire being a distant memory, Frodo will still get to be with his best friend in his new home.