What does Acts 13:41 mean?

Paul gave countless sermons in synagogues all over the Roman Empire (Acts 14:1; 17:17; 19:8); this completes the only transcription Luke gives us. Paul has been talking about how God literally saved the nation of Israel throughout their history, and how He has now sent Jesus to save them from their sins. He leaves them with this warning.

This verse quotes from Habakkuk 1:5: "Look among the nations, and see; wonder and be astounded. For I am doing a work in your days that you would not believe if told." Paul is, again, quoting the Septuagint which has slightly different wording.

Habakkuk is unique among the prophets in that the book is a back-and-forth exchange between the prophet and God. In Habakkuk 1:2–4, Habakkuk complains to God of all the injustice, destruction, and violence committed by the Jews in Judah. In Habakkuk 1:5–11, God says He will bring the Chaldeans—the Babylonians—to punish Judah. Habakkuk is startled by this information and asks how his nation can be punished by another that is even more wicked (Habakkuk 1:12—2:1). God tells Habakkuk to be patient, the Chaldeans will come to ruin (Habakkuk 2:2–20). In response, Habakkuk declares his faith in God (Habakkuk 3).

It's not clear, here, in what context Paul is using the verse. Certainly, there is a direct correlation in that within a few short decades Rome will destroy Jerusalem and the temple even more completely than Nebuchadnezzar had. And, like the Babylonians, Rome will eventually fall.

But Paul's emphasis is on God's saving work through Jesus Christ. It's possible Paul is just quoting Habakkuk as a general warning to listen and accept the message God has sent them. Some of the Jews and many of the Gentiles will, but the Jewish establishment, as usual, values their worldly influence more than spiritual salvation (Acts 13:42–52).