Okay, I don’t know that much about sports, but I’m going to assume this sentence is talking about sports. It means when good players show real skill and talent in a particular sport (when they play in their home countries), they are sometimes approached and recruited (or “snapped up”) by scouts or people who are representatives of wealthy European teams.
For example, let’s say the sport is soccer (or football), and there is a very good player who plays soccer very well for his local club in his home country. Wealthy European soccer teams can take notice of this player, and they will try to recruit the player so that he will play for their team. If they succeed, then people will say the player has been snapped up by the European team after he made his mark at home (meaning, his home country).