Supercompilation can be approached as a generalization of partial evaluation. The idea behind partial evaluation is that many parts of a program can be evaluated at compile time, and so should be. Supercompilation extends this, evaluating things that can’t be fully done at compile time as well, like turning map f (map g xs) into map (f . g) xs without anything besides the definition of map (At least I think I got partial evaluation right – I’ve only read much about supercompilation).
Another way to look at it is as a combination of many other optimizations, like deforestation, specialization, and inlining. By acting as if it already knew the inputs to functions and evaluating, it can get a more direct method of calculating the result – it can get rid of intermediate data structures by seeing how they will be used or it can plug in all possible values and then wrap the result in a case, or do something else with its pretend values.
Max Bolingbroke has a number of useful papers on the subject – I recommend the first one, Supercompilation by Evaluation, as an introduction. Section 2 introduces the subject by example and the rest, while a bit difficult to get through, is very informative about the process. Neil Mitchell also has a number of good presentations describing it.
I hope that helps.