Why do hidden Lego pieces have different colours?
Using different colors on the inside makes the instructions easier to follow. For example, in the case of the Statue of Liberty that you mentioned, if the entire model was sand-green, it would be easier to make mistakes in 1.) following the printed instructions, 2.) finding the pieces in the pile of loose elements, and 3.) attaching the pieces to the model under construction. 

Remember that LEGO sets are primarily designed for children, so LEGO puts a lot of testing and effort into making the instructions frustration-free. Using different colors is beneficial even for older LEGO fans who may start to have failing eyesight and would have difficulty finding pieces and seeing the instructions clearly. 

As far as which colors are chosen for the interior, they are something that contrast with the main colors of the model, easy to see in the instructions booklet, and easy to find in the pile of loose elements. Another thing that may be taken into consideration is that if the set has instructions for alternate models and what colors those alternate models would need (i.e. LEGO Creator 3-in-1 sets).