Acts 15:15 "And to this agree the words of the prophets; as it is written,"
The prophets declared this hundreds of years prior.
Acts 15:16 " `After this I will return, and will build again the tabernacle of David, which is fallen down; and I will build again the ruins thereof, and I will set it up:"
James is quoting from Amos 9:11, 12; "In that day will I raise up the tabernacle of David that is fallen, and close up the breaches thereof; and I will raise up his ruins, and I will build it as in the days of old:"
Amos 9:12 "That they may possess the remnant of Edom, and of all the heathen, which are called by My name, saith the Lord That doeth this."
That remnant of Edom, and all the heathen nations which will be called out, are being called out today. They are taking a new name, and that name is Christian, after "My Name, saith the Lord that doeth this." God is using men and women to preach and teach to Gentiles, and when they hear the voice of the Holy Spirit talking to them, they believe.
God tells us in Amos 9:9; "For, lo, I will command, and I will sift the house of Israel among all nations, like as corn is sifted in a sieve, yet shall not the least grain fall upon the earth." It doesn't matter where these people of the house of Israel are, or even those to be grafted in, God knows where each of them are, and he will sift all nations to find his own, those that will believe on the name of Jesus Christ, and by faith repent of their sins and follow His ways, shall find salvation.
Acts 15:17 "That the residue of men might seek after the Lord, and all the Gentiles, upon whom My name is called,' saith the Lord, `Who doeth all these things.' "
It is the wish of the Lord that not only the men of Israel, and Adomites take His name, but people of all races and nations will come to seek after the Lord. Those that call on the name of the Lord, call on Jesus' name, and they are the ones that shall be saved. James is quoting again here from the Old Testament, back in Amos 9:12.
Acts 15:18 "Known unto God are all His works from the beginning of the world."