The ancient Assyrian Empire fell apart because it was located in a very tough region of the world.  There were many other powerful empires in the region that were constantly fighting with one another.  The Assyrians took their turn as the winner in the struggles, but after a few centuries, they went back to being a loser and their kingdom was absorbed by the neo-Babylonian empire.

In Mesopotamia, there were a number of powerful contenders for dominance.  These included the Assyrians, the Babylonians, and the Hittites.  These groups took turns dominating the others.

While Assyria had its turn in power, it overextended itself like many empires do.  It got to be so big that the nobles who ruled the various parts of it started to see themselves as kings in their own right and would conduct wars of their own.  In addition, the brutality of Assyrian methods helped cause many people to hate the Assyrians and want to rise up against them.  These factors led to a weakening of the empire until it was finally defeated by Medes and Chaldeans in 612 BCE.  A few years after that, Assyria was part of the neo-Babylonian empire.