Why did many cities of the Persian state enthusiastically welcome Alexander the Great?

Because they greeted the Macedonian as a liberator. In fact, Persia was an empire that conquered many peoples and tribes. Naturally, this state of affairs gave rise to such an effect that when the Greco-Macedonian troops arrived, they were considered liberators from the rule of the Achaemenids. True, the empire of Alexander itself did not last long, since soon after his death civil strife began.