You understand ‘the children,’ identified in this verse or as ‘the sons of the kingdom’ in this verse, based just solely on the surrounding context. They are part of the original children, the unbelieving Judahites/Jews. This is supported by its contrast with the believing children or sons of the Kingdom, such as the Centurion as an example, and the many (far more than those in Israel) coming from the West and the East outside Judea of Israel. They are understood to be the true and faithful children.But the children of the kingdom shall be cast out into outer darkness: there shall be weeping and gnashing of teeth.” Matt 8:12
Mat 8:8 The centurion answered, “Lord, I am not worthy to have You come under my roof. But just say the word, and my servant will be healed.
Mat 8:9 For I myself am a man under authority, with soldiers under me. I tell one to go, and he goes; and another to come, and he comes. I tell my servant to do something, and he does it.”
Mat 8:10 When Jesus heard this, He marveled and said to those following Him, “Truly I tell you, I have not found anyone in Israel with such great faith.
Mat 8:11 I say to you that many will come from the east and the west to share the banquet with Abraham, Isaac, and Jacob in the kingdom of heaven.
Mat 8:12 But the sons of the kingdom will be thrown into the outer darkness, where there will be weeping and gnashing of teeth.”