How did ideas, like the worship of Isis, spread in the Ancient World?
Even though transportation was slower in antiquity and the present, and there was no instantaneous communication via the internet, people did travel widely and exchange ideas. There were two main forms by which religions were diffused across ancient regions.
The first method of religious diffusion was conquest. Egypt was, at many periods, a sprawling empire, exporting some of its religious practices to regions it conquered. When first Alexander the Great and then the Romans conquered Egypt, beliefs in Isis and other Egyptian cults were brought back home with soldiers and captives (especially slaves, who were usually taken from conquered nations).
The next method of religious diffusion was travel and migration, either for purposes of trade or sheer curiosity. Travelers to Egypt would learn of Isis and perhaps bring elements of her cult to their home countries, and migrants and travelers would establish temples where they stayed or settled.
Finally, it should be noted that unlike monotheistic religions, polytheistic ones are not exclusive. In other words, if you are a Christian you cannot also be a Muslim or a Hindu, but a polytheist normally acknowledged that many different deities existed, and participated in the cults of those deities of particular interest.
For example, a weaver living in Greece would worship Athena (patron goddess of weaving), as well as the other Olympian gods, and any local cults. If that weaver moved to Egypt, where the Egyptian gods reigned, that weaver would then also start participating in the local cults of Isis and other Egyptian gods. If the weaver then became wealthy and successful in Egypt after making sacrifices to Isis, then when the weaver retired home to Greece, that weaver might erect a local shrine honoring Isis in thanks for her favor.
Finally, the Ptolemaic dynasty promoted the worship of Isis as a way of synthesizing Hellenistic and Egyptian religious traditions, and thus some of the expansion of the cult in the Hellenistic period was due to state subsidies.