What does break the floodgates of countless uncried tears mean

A floodgate is what it sounds like - a gate that holds back dangerous waters from something that needs to be protected; typically some sort of shoreline civilization.  To break it sets the water forth, which is represented as tears here.  This basically means that whatever was stopping that character from crying, whether it be pride, the situation, or other factors, is now gone, and they're ready to let out all of their pent-up tears.

- "chilles is sad and angry because hector, the trojan leader, killed his best friend patroclus. he seems angry with himself because he didn't or protect patroclus against the trojans."