Repentance - Sorrow for sin and forsaking of it. It was proper that the “necessity” of repentance should be preached among all nations, for all were sinners. See Acts 17:30.
Remission of sins - Pardon or forgiveness of sins. It should be proclaimed that all people should repent, and that those who are penitent may be pardoned.
In my name - By my command it should be proclaimed that people should repent, and by my merit that they may be pardoned. Pardon is offered by the authority of Christ to all nations, and this is a sufficient warrant to offer the gospel “to every man.”
Beginning at Jerusalem - This was the dwelling of his murderers, and it shows his readiness to forgive the vilest sinners. It was the holy place of the temple, the habitation of God, the place of the solemnities of the ancient dispensation, and it was proper that pardon should be first proclaimed there. This was done - the gospel was first preached there. See Acts 2. Paul also, in his travels, preached the gospel “first” to the Jews, the ancient people of God, offering them pardon through their own Messiah; and, when “they” rejected it, turned to the Gentiles, Acts 13:46.