As with Cesc Fabregas, some players who go and play for foreign clubs improve on a cultural level. It makes them grow on many levels; intellectually, because you have to learn a new language and adapt to another culture, and on a footballing level too.
I was lucky enough to attend schools where they were understanding about when I needed to go abroad to play chess. Of course, socially it is important to go to school and interact with people your own age.
I think self-awareness is probably the most important thing towards being a champion.