When we come in the name of the Lord--not in our own names--people who reject us are rejecting the peace of the kingdom. Our hearts should weep over this.
God extends to the world His hand of mercy, forgiveness and reconciliation through the cross of Jesus Christ. If people refuse His gift, eventually judgment and condemnation will come upon them (see Luke 10:16).
When we are sent by God, we become Christ's voice, hands and feet. We are His ambassadors, His representatives. This is why we must be filled with His praises, His peace, His wisdom and His strength.
Wherever you go, speak the very words of God. Speak as if God Himself were speaking through you. As you develop a deeply intimate relationship with the Lord, His Word will flow out from your relationship with Him to the world.
It's not just the words you speak that will turn people to the Lord. It will be the power of the Holy Spirit that flows from your innermost being through your words that will bring life to others.