The Man with the Braces is a host on the reality TV show Frame Up.
When Homer and Marge appeared on the show and Myrna Bellamy's murder was staged, the Man with Braces "stole" her necklace and made sure that Marge and Homer saw him leaving the house with it. This made Marge and Homer, who were already the prime suspects, look even more guilty because they were accusing a mysterious third party of the crime ("It was the man with the braces!") without having any evidence to back them up.
When all was revealed on the show, the real Man with Braces identified himself as "some actor they hired." His real name isn't known.