First, I recommend keeping an example in mind. For instance, let C be a set of cars, R1 = {(c1, c2) | c1 and c2 have the same model (e.g., Accord)} and R2 = {(c1, c2) | c1 and c2 have the same make (e.g., Honda)}. It is assumed that different manufacturers produced different models, so if c1 and c2 have the same model ((c1, c2) in R1), they also have the same manufacturer ((c1, c2) in R2), i.e., R1 is a subset of R2.

As you write, the fact that is a refinement of means that every set in is a subset of one of the sets in -- this is what we need to prove. Fix a set . This set is an equivalence class consisting, for example, of all Accords. Take some . Then has an equivalence class with respect to , consisting of all cars with the same make as . We'd like to show that . Well, take any , i.e., , i.e., is another Accord. Since all Accords have the same manufacturer ( ), , i.e., .