Can anyone explain what happened to Radagast the Brown after the Hobbit? What was he doing after and did he stay in Middle Earth?

Radagast sent the Gwaihir to Isengard with a message for Gandalf. The Eagle found Gandalf trapped at the top of the tower and rescued him.

Radagast dwelt in a place called Rhosgobel, near the western eaves of Mirwood and not very far from Dol Guldur. When Elrond's scouts went to Rhosgobel in search of the Brown Wizard, they found the place abandoned (though we don't get to see or hear about this in the films). What had become of Radagast is not recorded.

Radagast was the one who told Gandalf that Saruman had sent for him. He may have sent Gwaihir to rescue Gandalf if he had learned about Saruman's treachery. but when Elrond's retainers looked for him in Rhosgobel, they found it abandoned, and nobody found him.

It was never mentioned what happened to Radagast. Whether he was killed or went into hiding is unknown.