Fame points is what you gain by defeating a human opponent.
You will get Fame points in two cases: if you beat someone during attack (Battle button on lower left-> Find an opponent) or if you were able to defend your base so attacker didn't gain 50% damage (didn't gain any stars).
Algorithm of Fame point' calculating is a complex function, but it boils down to how much stronger/weaker you than your opponent. Battle with weak opponents give you a small amount of fame (even 0 in some cases!), battle with strong opponent may give you a lot of fame.