Radagast The Brown disappeared from Arda after the first book The Fellowship of the Ring. What happened to him?
It is not known.
We do not know what he did during the whole duration War of the Ring or afterwards. Tolkien doesn't have any mention of him.
We do know that by mistake he had Gandalf captured by Saruman
By the time of the War of the Ring Radagast did not dwell any more in Rhosgobel. In summer T.A. 3018 Saruman told Radagast that he was willing to help Gandalf, and sent the Brown Wizard to seek him out at once. Radagast did not know much of Eriador but sought for the Shire, knowing that he would find Gandalf nearby. [...] Radagast warned Gandalf that the Nazgûl were abroad, disguised as riders in black, and that they were seeking news of the Shire. He also gave him Saruman's invitation [...] With that he rode away back towards Mirkwood.
By sending Gandalf to Orthanc, Radagast unwittingly had him captured. Saruman's message proved to be a trap for Gandalf who was imprisoned in Orthanc, but still he did not believe that Radagast was also a part of Saruman's plans. Indeed, it was thanks to Radagast that Gandalf was able to escape from the pinnacle of Orthanc upon the wings of Gwaihir.
-- The Lord of the Rings, The Fellowship of the Ring, "The Council of Elrond"
Radagast's actions during and after the rest of the War are not recorded. After the Council of Elrond, many scouts were sent out from Rivendell to many different locations. Some passed over the Misty Mountains and eventually came to Rhosgobel, but they found that Radagast was not there.
-- The Lord of the Rings, The Fellowship of the Ring, "The Ring Goes South"
Apart from what Shevliaskovic said (and what is absolutely true), it's noteworthy to add this:
0. Radagast is quite similar to the two "unknown" Istari (so-called Blue Wizards) because he handled "background matters" (general maintenance, I'd say) instead of direct action. Thus, we don't know much about them at all, and their fate after The War of The Ring is largely unknown.
1. Radagast was initially the man Isengard was going to be given over after Sauron has fallen:
Gandalf looks at him. 'I am the White Wizard now,' he said - 'look at your many colours.' Saruman is [?clad] in a filthy mud colour. 'They seem to have run.' Gandalf takes his staff and breaks it over his knee. [?He gives a thin shriek.] 'Go, Saruman,' he said, 'and beg from the charitable for a day's digging.' Isengard is given to the Dwarves. Or to Radagast?]
From: The Treason of Isengard, p. 212, from: The History of Middle-earth, by JRR Tolkien, collected by Chistopher Tolkien
2. We aren't sure if Radagast actually succeeded or failed, and thus what happened to him; Tolkien wrote that he gave up his mission as one of the Wizards by becoming too obsessed with animals and plants. Still, since he was specifically chosen by Yavanna, while he failed at helping with dealing with Sauron and his minions, mainly due to exposing Gandalf to Saruman, he may have been also/mainly assigned to protect the flora and fauna of Middle-earth, a task that would not end with the defeat of Sauron and the end of the War of the Ring - and one at which he, arguably, succeeded.
Indeed, of all the Istari, one only remained faithful, and he was the last-comer. For Radagast, the fourth, became enamoured of the many beasts and birds that dwelt in Middle-earth, and forsook Elves and Men, and spent his days among the wild creatures. Thus he got his name (which is in the tongue of Numenor of old, and signifies, it is said, "tender of beasts").
The essay on the Istari just cited thus tells much about them and their origin that does not appear in The Lord of the Rings (and also contains some incidental remarks of great interest about Valar, their continuing concern for Middle-Earth, and their recognition of ancient error, which cannot be discussed here). Most notable are the description of the Istari as "members of their own high order" (the order of Valar), and the statements about their physical embodiment. 5 But also to be remarked are the coming of the Istari to Middle-earth at different times; Círdan's perception that Gandalf was the greatest of them; Saruman's knowledge that Gandalf Ring, and his jealousy; the view taken of Radagast, that he did not remain faithful to his mission; the two other "Blue Wizards," unnamed, who passed with Saruman into the East, but unlike him never returned into the Westlands; the number of the order of the Istari (said here to be unknown, though "the chiefs" of those that came to the North of Middle-earth were five); the explanation of the names Gandalf and Radagast; and the Sindarin word ithron, plural ithryn.
The note ends with the statement that Curumo [Saruman] took Aiwendil [Radagast] because Yavanna begged him, and that Alatar took Pallando as a friend.
There is no hint of an explanation of why Yavanna's evident desire that the Istari should include in their number one with particular love of the things of her making could only be achieved by imposing Radagast's company on Saruman; while the suggestion in the essay on the Istari (p.407) that in becoming enamoured of the wild creatures of Middle-earth Radagast neglected the purpose for which he was sent if perhaps not perfectly in accord with the idea of his being specially chosen by Yavanna. Moreover both in the essay on the Istari and in Of the Rings of Power Saruman came first and he came alone. On the other hand it is possible to see a hint of the story of Radagast's unwelcome company in Saruman's extreme scorn for him, as related by Gandalf to the Council of Elrond: "Radagast the Brown!' laughed Saruman, and he no longer concealed his scorn. 'Radagast the Birdtamer! Radagast the Simple! Radagast the Fool! Yet he had just the wit to play the part that I set him.'" Whereas in the essay on the Istari it is said that the two who passed into the East had no names save Ithryn Luin "the Blue Wizards" (meaning of course that they had no names in the West of Middle-earth), here they are named, as Alatar and Pallando, and are associated with Oromë, though no hint is given of the reason for this relationship. It might be (though this is the merest guess) that Oromë of all Valar had the greatest knowledge of the further parts of Middle-earth, and that the Blue Wizards were destined to journey in those regions and to remain there.
From: Unfinished Tales, II. Istari.
As such, we can safely assume he stayed in ME in the Eras to come to fulfill Yavanna's desire for him to watch over creatures of ME - see Radegast; seems old Raddy was still around not so long ago chuckle.
Additional reading here (quite a good essay, so to speak).