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