What does Acts 10:28 mean?

Peter's statement is harsh, but it is a matter of Jewish survival. Shortly after entering the Promised Land, right after Joshua died, the Israelites started worshiping the gods of the people God meant for them to displace (Judges 2:1–5). This idolatry continued until the nation split into the northern kingdom of Israel and the southern kingdom of Judah—and then it got worse. About 680 years after the Israelites entered the Promised Land, Israel was all but demolished by the Assyrians. About 120 years later, Judah was taken into exile in Babylon. Fortunately, Judah returned, largely intact as a people.

After a few fits and starts, the Jews resolved to serve God alone. God had told them to be a people set apart, noted for their obedience to Him through the Mosaic law. In their zeal, the Jews added to the Law and fought off their foreign oppressors whenever possible. When they couldn't, they made it clear they would not associate with unclean Gentiles. This is why Jesus' accusers remained outside Pilate's house—so they wouldn't get unclean and be ineligible for the Feast of Unleavened Bread (John 18:28), and why the Pharisees scolded the disciples for not washing their hands after leaving the marketplace—who knows where that food had been! (Mark 7:1–5)

It is true that God wanted His people to be apart from the pagan nations and their idol worship, but only for a time, and only so far as it involved those sins. God promised Abraham that all the world would be blessed by his descendants (Genesis 12:2–3), and this can't happen without interaction. Peter begins to understand this now, and he will defend his actions before the church leaders in Jerusalem (Acts 11:1–18). But he will lose sight of it later when the church leaders have opportunity to witness Peter's actions (Galatians 2:11–14).