Trophies are chicken number milestones for each egg. They are unlocked by completing all missions.
Trophies are gained when the number of chickens on your farm reaches a certain level.
10 million for Bronze Trophy
50 million for Silver Trophy
250 million for Gold Trophy
1 billion for Platinum Trophy
10 billion for Diamond Trophy
Since it is harder to get each amount of chickens with lower value eggs, trophies for lower value eggs are harder to obtain and typically offer better rewards.
The game always keeps track of the most chickens you've had on a home farm for each type of egg, but doesn't award any trophies until all missions are complete. Once you complete the final mission (Lay 250T Dilithium Eggs), the game will award you with all trophies that you would have otherwise acquired earlier.