The type of hyperspectral camera [0] that they implement is a push broom scanner [1] which works like this: the data is collected line for line of an image of the scene, e.g. by a rotating mirror that changes the view all the time. A narrow slit then selects a line out of this view. The light from that line is then passed through a dispersive element (optical grating, prism) in such a way that a spectrum is generated perpendicular to the scanned line. That means that the image generated on the sensor of the camera now has the spectrum of each pixel of the scanned line distributed over its different rows. Therefore reading out all of the sensor gives the full spectrum for that line. Since the mirror is rotating, the line is scanned over the scene and collecting the data from frame to frame eventually results in a hyperspectral cube instead of a normal image. Since the data is collected over many frames, this method is trading temporal resolution/frame rate for spectral information.
[0] https://en.wikipedia.org/wiki/Hyperspectral_imaging#Spatial_...