Why did Albus Dumbledore put on the ring?

Answer 1

Albus Dumbledore was considered a great wizard and many wanted him to be the Minister for Magic - a job Dumbledore didn't trust himself to have. Instead, he was happy being the headmaster of Hogwarts, a important role in the British Wizarding community.

Answer 2

Albus Dumbledore lost both his father, mother and sister at a young age. He experienced a lot of death in his life and when he found the ring which contained the resurrection stone he thought that wearing the ring would help him see his family. Dumbledore was so desprate to see them that he didn't think about what the Horcrux would do to him.