In order to achieve the effect that Claude Rains wasn't there when his character took off the bandages, the director had Rains dressed completely in black velvet and filmed him in front of a black velvet background.
On the DVD short documentary, Claude Rains' daughter tells of a time when the two went to see this movie in the theater years after it was made. It was bitterly cold and his face was completely covered by a hat and scarf. When he spoke to ask for the tickets, the attendant immediately recognized his voice and wanted to let them in for free. Rains was quite upset at this and demanded that he pay full price.
When screenwriter R.C. Sherriff came to Hollywood to write The Invisible Man (1933), he asked the staff at Universal for a copy of the H.G. Wells novel he was supposed to be adapting. They didn't have one; all they had were 14 "treatments" done by previous writers on the project, including one set in Czarist Russia and one set on Mars. Sherriff eventually found a copy of the novel in a secondhand bookstore, read it, thought it would make an excellent picture as it stood, and wrote a script that (unlike the Universal versions of Dracula (1931) and Frankenstein (1931)) was a closer adaptation of the book.
According to information given on TCM before the movie is shown, Claude Rains was also chosen for the role because he spoke with such clarity and could be easily understood. This was important since he was 'invisible' or his face was covered for almost the entire film.
The trivia items below may give away important plot points.
Boris Karloff had been the studio's original choice for the role of the Invisible Man. He turned down the role because he would not be seen on screen until the end. Director James Whale wanted someone with more of an "intellectual" voice than Karloff. He selected Claude Rains after accidentally hearing Rains's screen test being played in another room. (Until this film, Rains had primarily been a stage actor. Although he had appeared in one silent movie (Build Thy House (1920)), this was his first sound film.)