to proclaim publicly like a herald: to announce a message openly and with authority, on behalf of the one who sent you
In the ancient world a herald was the official who stood in a public place and announced what a king, a city, or a god had decided. The crucial thing about him is that the message was never his own. He did not invent it or argue for it; he delivered it, loudly and plainly, on someone else's authority. That picture sits underneath this word every time the New Testament uses it, which is why Paul can say the point of his work is that "we do not announce ourselves, but Jesus Christ as Lord."
Most often the message being announced is God's. John the Baptist and then Jesus proclaim that God's kingdom has arrived and that people should turn back to God, and after the resurrection the apostles proclaim Jesus himself, crucified and risen, as Lord. The word names the act of announcing, not the skill of persuading, which is why it can stand alone as a shorthand for the whole work of spreading the Christian message. It also carries a sense of being sent: to proclaim in this way, someone must first commission you, as Paul makes plain when he asks how anyone can herald a message unless they are sent to do it.
There is a further weight to the word that plain "announce" can miss. The herald's announcement is not just a report of something that happened elsewhere; it is the public event through which the thing announced actually arrives. When the kingdom is proclaimed, the proclaiming is part of how the kingdom comes near and forces a decision.
The same word also covers ordinary public announcing. People whom Jesus had healed could not keep quiet and proclaimed everywhere what had happened to them, and an angel in Revelation proclaims a challenge in a loud voice. Whether the content is the gospel or simply astonishing news, the action is the same: making something openly and audibly known to everyone within reach.