to declare something with solemn, God-invoking weight: to testify to the truth, or to lay a serious charge on someone
This word is a strengthened form of the ordinary verb for bearing witness (μαρτυρέω). The added prefix pushes it past simply reporting a fact to affirming it with full solemnity, the way a person speaks when God is called on as witness. Its force lies not in the content of the speech but in its seriousness.
That solemn weight points in two directions. When it is aimed at the truth, it means to testify or bear witness, and Luke makes it almost a technical word for the apostles' public witness to the gospel, the kingdom of God, and the risen Christ. When it is aimed at a person, the same solemnity becomes a charge or a warning, a command laid on someone before God, which is how Paul uses it when writing to Timothy.
The two uses are one act seen from two sides: a speaker staking everything on the gravity of what is said. Related words divide up work that English needs several words for. Another verb, μαρτύρομαι, also calls someone to witness, while παραγγέλλω carries the plain force of an order without the note of bearing witness.