How is the reflectance spectrum of solids measured?
As the name implies, reflectance spectra are measured from light reflected off an object. Any object with reflects light, such as any opaque object) will work for this technique; to me this seems like solids are ideally suited for reflectance spectrphotometry. You might be used to transmission or absorbance spectrophotometry, which measures the amount of light transmitted or lost after passing through a sample. These methods are typically used for liquids which allow passage of light to some extent. 

The paper (1) cited by the review you mention says that they used a "RAMSES-ARC hyperspectral radiance sensor," which is probably just as awesome as it sounds. In principle reflection spectrophotometry works very similarly to transmission-based spectrophotometry: generate monochromatic light, shine it at the sample, and measure the intensity of light that comes back at that wavelength (or over a range of wavelengths)$^†$. If you do this for many different wavelengths of light, you have a reflectance spectrum. This link might have some useful information.

$^†$ Edit: Modern-day spectrophotometers use a more efficient method in which broad-band light is passed onto the sample and the response at a single wavelength can later be mathematically reconstructed. If you're interested in these types of details, refer to Fourier Transform spectroscopy.



(1): Doring TF, Anchetti M, Hardie J. (2009). Autumn leaves seen through herbivore eyes. Proc. R. Soc. B 282(1801): 121-127.