Question

how do scientists develop scientific explanations about subjects that are impossible to study through controlled experiments?

Answer 1

There are many ways to do this. You can either create a close approximation to what you think are the most relevant variables and hope your idea about the relevant variables was true.

Another way would be to do different kinds of experiments and hope you can deduce information from those experiments onto the subject that you would like to find an explanation for. 

Answer 2

By following the steps of the scientific meathod