A Lytro-style light-field (plenoptic) camera still uses a normal main lens and an image sensor, but it adds a microlens array just in front of the sensor.

In a conventional camera, each sensor pixel mainly records how much light lands at one position on the sensor. In a light-field camera, each microlens forms a tiny partial image on a group of sensor pixels. That lets the camera record not just where light hit, but also something about the direction the light came from.

A useful analogy is a grid of many tiny cameras shooting at once from slightly different viewpoints. Software then combines those many slightly different views to reconstruct images focused at different depths, and can enable limited viewpoint/perspective changes.

So it is not capturing an infinite amount of information. It samples the light field at limited spatial and angular resolution. The tradeoff is that you gain directional information, but final image resolution is lower than using the whole sensor as one flat image.

In short: traditional camera = one image focused on the sensor; light-field camera = main lens + microlens array + computation to reinterpret the recorded rays after capture.