What does Matthew 10:19 mean?

This core group of disciples (Matthew 10:5–6), sent out on Christ's behalf, are being warned they will eventually face great persecution for His sake. Most of these trials will come after Jesus' death, resurrection, and return to heaven. That mistreatment will come from every level of authority in their lives, both Jewish and Gentile. It will include trials before religious Jewish authorities, secular Jewish leaders, and Roman/Gentile authorities.

Encouragement, in this case, comes from Jesus telling His disciples that persecution is not a failure of God to protect them. Instead, this is exactly God's plan. When they are "dragged" before these courts, they will have the opportunity to tell all these powerful people about the gospel of Jesus, about salvation through faith in Christ. This is what God intends to happen. Paul, notably, will take advantage of this very idea when he is brought before a Roman court (Acts 26:1–3).

In prior passages, Jesus instructed this group of men not to make material preparations for their journey (Matthew 10:9–10). Here, He gives reassurance that they ought not be worried about what to say when they are confronted. This verse does not explicitly forbid the disciples from thinking about what they will say in those moments. However, the tone of Jesus' comments strongly suggests He wants the Twelve to simply leave those moments in the hands of the Holy Spirit (Matthew 10:20). Rather than worrying (Matthew 6:34), they should trust that proper words will come to them when needed.