What does Acts 8:35 mean?

The passage the Ethiopian court official is reading is Isaiah 53:7–8. It describes the Suffering Servant's unjust treatment and death. The Ethiopian must have been surprised to learn that this is good news. It's because of Jesus' death on the cross that he—and all of us—can be reconciled to God. He responds immediately and insists on being baptized (Acts 8:36).

Philip's explanation follows the common format of evangelism in the early church to Jews: he explains how Jesus of Nazareth fits into the Jewish Scripture (our Old Testament). Peter did the same on the day of Pentecost when thousands of Jews flooded Jerusalem for the feast and discovered the Jesus-followers whom the Holy Spirit had just indwelt (Acts 2). Paul will continue the pattern; in each town he goes to visit, he starts either with the synagogue (Acts 9:20; 13:14–49), or another place where Jews meet (Acts 16:13). Jesus started this tradition when He met with the two men on the road to Emmaus: "And beginning with Moses and all the Prophets, he interpreted to them in all the Scriptures the things concerning himself" (Luke 24:27).

That doesn't mean that we should always start with the Old Testament when introducing people to Jesus. As Jesus did during His ministry, we should start by meeting the person where they are. Paul did this when he spoke with the Greek philosophers in Athens. He began by putting their monument to an unknown god into context, and then quoted their own philosophers to show that God has sovereignty over His creation. Finally, he explained that God's sovereignty extended to His right to judge them and to raise the dead (Acts 17:22–31).