Why do LDR photocells use CdS and CdSe?
What makes cadmium selenide and cadmium sulfide such appealing
  semiconductors to make photocells out of?


Cadmium sulfide and Cadmium selenide turn conductive in the visible wavelengths, they do this because they are semiconductors and the bandgap is responsive to visible wavelengths. 


  A photoresistor is made of a high resistance semiconductor. In the
  dark, a photoresistor can have a resistance as high as several megohms
  (MΩ), while in the light, a photoresistor can have a resistance as low
  as a few hundred ohms. If incident light on a photoresistor exceeds a
  certain frequency, photons absorbed by the semiconductor give bound
  electrons enough energy to jump into the conduction band. The
  resulting free electrons (and their hole partners) conduct
  electricity, thereby lowering resistance. The resistance range and
  sensitivity of a photoresistor can substantially differ among
  dissimilar devices. Moreover, unique photoresistors may react
  substantially differently to photons within certain wavelength bands.
  Source: Wikipedia Photoresistor



Source: Optical and photoelectrical properties of CdSxSe1-x films produced by screen-printing technology


  But other direct bandgap semiconductors exist, too, including ones
  without any highly toxic elements, like gallium antimonide or tin
  sulfide, or any of the more complex compounds used for thin-film solar
  cells. What gives cadmium compounds such an advantage?


There are other materials that are responsive, but CdS or CdSe have responsivity in the visible wavelength range, this is useful if you want your device to respond in the same wavelength range that people see in. CdS is the best photosensitive material in the visible range and photodetectors made from CdS are cheap. The alternative would be a photo diode circuit, which is more complex and requires more components. 


Source: http://www.resistorguide.com/photoresistor/