The Eyes are the true antagonists of the Don't Hug Me I'm Scared webseries. The eyes represent the media, and how the media controls society.
The eyes were briefly shown during the creative explosion scene of the first video. While the three puppets were busy being creative, the camera proceeded to pan out, and it revealed that the show was being made by Hollywood. The Eyes are also shown as being a part of the stage devices. One could also say that the scene in which the Notepad places a monocle over its one eye references the eyes.
The eyes are also frequently seen in the sequel video, TIME. One scene in which the Eyes appear is when the three puppets were strapped into chairs, and are forced by Tony to watch the computer screen.
The Eyes represent the media and how the media conditions its young audience to think a certain way. The Eyes could also be a reference to Big Brother, the antagonist of the 1949 Orwell novel 1984.