What does Acts 17:15 mean?

Paul is used to being run out of town (Acts 13:50–51; 14:5–6, 19–20; 16:35; 17:10). What's new for him is having a large enough team to delegate responsibilities when that happens. He already left Luke in Philippi, as evidenced by the change in pronouns from "we" to "they" (Acts 17:1). Now, escaping the Thessalonians by running to Athens, Paul leaves Timothy and Silas behind in Berea.

In Athens, Paul will bring a "new teaching" (Acts 17:19) to the Stoics and Epicureans. The philosophers will invite him to speak at the Areopagus to fulfill their desire for novel ideas (Acts 17:21).

Paul will tailor his message to his audience, using poetry to introduce the Jewish God as the Creator of all. The Athenians follow along until Paul mentions God raised His appointed One from the dead; the philosophers don't believe anything survives after death, so resurrection sounds rather primitive (Acts 17:22–32).

Meanwhile, Paul changes his mind about Timothy and Silas joining him. He sends Timothy back to check on the church in Thessalonica (1 Thessalonians 3:1–2, 6) and apparently Silas either stays in Berea or goes somewhere else in Macedonia. They don't catch up with Paul again until he has been in Corinth for some time (Acts 18:5).

The traditional understanding of this verse is that Paul took a ship down the eastern coast of Macedonia and Greece. Another reading of the text suggests the church in Berea pretended to take Paul to the harbor but once the Thessalonians were tricked, walked him to Athens.