What happened to Alexander's empire after he died and why?

Answer 1

He had no clear heir, so his Macedonian generals took the provinces for themselves, declared themselves kings and fought amongst each other.

These we call the Hellenistic Kingdoms because they introduced a veneer of Greek government and culture within them.

Answer 2

He left no clear successor, and his generals seized various parts of the empire for themselves and fought over the spoils, eventually forming several kingdoms which we today call the Hellenistic Kingdoms, most durable of which were Egypt (by Ptolemy), Syria (by Seleucis) and Macedonia.