What is shutter speed? Faster than light?

A camera cannot capture an image before light gets in. I think you misunderstand what she said, or she doesn't know what she is saying.

The ony shutter speed that is faster than light is a shutter that never opens.

The MIT camera is not faster than light. It is just a very, very fast camera with a very fast shutter speed. This lets in only a tiny bit of light.

Shutter speed is a measurement of time that light is allowed to strike either the sensor in digital cameras or the film in analog cameras. It is one of two factors that affect exposure directly, the other being aperture of the lens which determines how much light can pass through the lens. ISO, commonly though of as an 'exposure setting', actually doesn't figure into exposure in it's most strict sense. It is a measurement of relative sensitivity of the medium to be exposed, the film or the sensor to light. Commonly, shutter speeds can be as fast on modern DSLR cameras as 1/8000 of a second. Or as slow as several minutes if 'bulb' setting is used. Even 1/8000 of a second isn't faster than light, if it was, there would be no image captured. In practice, however, if one leaves the lens cap on, the 'image' will be all black. In digital terms, the cleared state of the sensor is recorded as the image, meaning all black. In film terms, the developed frame will be either clear for a negative print film, leading to an all black print or all black for a positive slide film leading to all black slide. These aren't truly an image capture in any case, since no sensitivity at all was recorded. They are merely the relative 'blank' states of the media, like watching TV with no signal input creates dead screens.