Other's probably have a more clear idea of what happened than me, but from what I gathered 2 things contributed to the fall of Pulse civilization.

1) After the war with cocoon, the people of pulse turned to war with each other for resources.

2) Fal'Cie started turning the people of Pulse into L'cie at an increasingly higher rate as time went on.

In result, the people either killed each other off or were turned to l'cie. That is what I gathered from the analects at least.