The black disc with holes is an occluder. It's used when someone can't read down the chart as far as people normally do. It's important for the screeners to know whether the problem is at the front of the eye or in the retina at the back of the eye. If the lens is too thick, light passing through the edges gets " bent" too much, making the letters blurred and difficult to read. When you look through the little holes, light enters your eye through the central part of your lens where it passes through without being "bent" so it doesn't matter if your lens is too thick, the letters will be in focus. If someone can't read any further down the chart when they use the pinhole, it suggests that there may be a problem within the retina.
Some people have trouble using the occluder so it's always important to use distance glasses, if you have them, and make sure your prescription is up to date.