Q:

Why is liquid fat transparent and solid fat opaque? 

While frying some chicken I was wondering why the fat becomes transparent when melting. 
As far as I know, transparency is caused by the fact that the substance can't absorb photons of visible light (due to band gap or other excitation mechanisms)
Is this the reason for liquid/solid fat as well?  How does the fat lose excitation possibilities when molten? 

A:

The same phenomenon occurs with ice/ water, butter, Coconut Oil etc.  
Let's look at butter.  
While it is a liquid, it is yellow in color.  This means that it is reflecting and transmitting only yellow light.  When it turns into solid, it still appears yellow, but in this case, it does not transmit any light, it just reflects it. 
This is because when the butter is solidified, certain small crystals ( Crystallites ) are formed which absorb all colors except yellow ( as expected  ) and they reflect all the yellow light back ( because their refractive index becomes very high as explained by XcoderX in the comments ).