Light field cameras measure not only the ray position on the sensor, but also the ray angle. I think of it as an alternative to taking a series of images of a static scene over a range of camera focal lengths or axial detector positions (time sequential imaging). All these approaches enable the photographer to get in-focus images over a range of object distances, thereby extending the depth of field.
This single-shot multiplexing is achieved with a microlens array in front of the sensor. An example I designed to illustrate the “Lytro” approach is shown below. The primary optic focuses the rays at the microlens array, which is located one microlens focal length from the sensor. Just as the large lens transforms a collimated field of rays into a focused spot, that focused spot is transformed by the microlens into a smaller collimated field of rays. Thus, this is called “light field imaging”.