For most of us, this phrase brings with it a question: What is the Gospel (good news) of the Kingdom? It is, sadly, an aspect of the Gospel we proclaim that we seem to miss...
In 1 Corinthians 15, Paul presents the message of the Gospel: That Christ came, died, and resurrected from the dead according to the scriptures (verses 1-23). Paul then continues to describe in this text the story of the rule of Christ (His Kingdom)
The highlighted verse is a quotation that Paul uses from Psalm 110:1. This is the good news of the Kingdom. Through the rule of the Messiah, God will put the world right again. He will bring all things to Himself.
Isaiah 2:3-4 paints a picture of the nations streaming up to the mountain of God
Isaiah 9 tells us that "El Gibor" (The Father of Eternity) is coming and He is bringing with Him a kingdom that will not end.
This was the promise given to the Israelite's! That yes, the Messiah was coming to bring Salvation, but that is not all that He would bring. The good news was that Jesus would come to sit and rule on the throne of David, that the whole world would belong to Him, and that he would bring the whole world to Himself! Look for a moment if you would at Daniel 7:13-14:
his dominion is an everlasting dominion, which shall not pass away,
and his kingdom one that shall not be destroyed.