As far as I understand, the RG theory, or functional RG theory is a mathematical tool for moving in the "scale dimension". The tool can be used for calculation of Feigenbaums constant (e.g. mentioned here). Can the theory be given a simple example of how to move one "step" in the "scale dimension" ?
When you look at the Mandelbrot set M, you can observe infinitely many "Baby Mandelbrot" copies of the set within M and so on. Renormalization allows you precisely to move from one step (the Baby copy) to the next one. For more details, see for example the book by McMullen "Complex dynamics and renormalization", or that article by M. Lyubich: http://smf4.emath.fr/Publications/Gazette/2007/113/smf_gazette_113_45-50.pdf