Most spectrometers work by spatially separating light into its component frequencies by means of a diffraction grating, and measuring that with a CCD array. That means measuring the intensity*, or power per unit area, the units of which are watts per square meter. However, there is usually an unknown scaling factor involved, so don't expect the values to actually be in W/m². I would guess that the values in these files are normalized so that a value of 1 is the intensity at which the corresponding CCD pixel saturates.
*Strictly speaking, you are measuring the intensity present in one range of frequencies, but it's simpler and usually justified to just treat it as one frequency.