What does Acts 5:5 mean?

Neither Peter nor the church had the civic authority to judge people and condemn them to death. Even Jesus was careful to only speak the words of His Father (John 14:10). Peter has extra reason to be circumspect in his judgment of Ananias. Satan has filled Ananias with rebellion against God (Acts 5:3). At one point, Peter allowed Satan to push him to rebuke Jesus (Mark 8:32–33). Only the indwelling of the Holy Spirit could make Peter pronounce such strong judgment, and it's obvious it is the Holy Spirit who kills Ananias.

It is consistent with the history of Israel for God to make a strong statement at the beginning stages of a great shift in His work. Before the Israelites were out of the desert, Aaron's sons Nadab and Abihu gave an offering to God that He did not endorse; God responded by consuming them with fire (Leviticus 10:1–2). As Joshua began to lead the Israelites into their inheritance, a man named Achan took spoils that God had claimed for Himself. God demanded Achan's execution (Joshua 7). When the Jews came back to Jerusalem from captivity in Babylon, many had married women from cultures God strictly forbade. They had just returned from a seventy-year punishment for worshiping foreign gods, and now they had wives who would lead them to do so again. God ordered them to separate from both their wives and their children in order to protect the re-establishment of godly Jewish culture (Ezra 9—10).

God explained His strong measures to Aaron after the death of his sons: "Among those who are near me I will be sanctified, and before all the people I will be glorified" (Leviticus 10:3). A strong beginning is essential for any kind of lasting change.