Can you explain, how one can understand the "sanctified"/"holy" in this verse, especially as it is used with regard to the unbelieving spouse and the children?
In this verse Paul is answering the issue that a Christian husband or wife might have if they are married to a non-Christian. The concern is something like this: "If I stay married to an unbeliever, won't I be spiritually defiled? Won't my union with Christ be compromised by physical union with someone who does not belong to Him?" This anxiety is understandable, especially in light of Paul's previous teaching about being united to Christ and maintaining one's body as the temple of the Holy Spirit.
Paul's answer to this concern is clear and concise: "the unbelieving husband has been sanctified through his wife," and the same applies vice versa. The verb translated "has been sanctified" is perfect tense, which means Paul is not describing a future conversion or even the beginning of one. The unbeliever remains an unbeliever. Rather, Paul is describing a status that was established at the moment the other spouse became a Christian. Because husband and wife are "one flesh," as Paul has taught in 1 Cor. 6:16, 17, the sanctification of the believing spouse necessarily extends to the marriage bond itself. In that sense, the unbelieving spouse becomes sanctified in relation to the marriage, not in relation to salvation.
What does this mean? When the believing spouse offers themselves to God, they cannot help but present their marriage to God as well. The believer now treats the marriage -- and therefore the spouse -- as belonging to God in a special, set‑apart manner. Whatever the spouse may be in themselves, in the believer's eyes they are now regarded as belonging to a sphere that is set apart to God. The same is true for everything that belongs to a Christian. The reason is that when God takes a man into covenant with Himself, He takes the whole of that man, all that he is and all that he has. His possessions, abilities, and relationships are all to be treated as holy because they are bound up with a life devoted to God.
This brings us to the children. Paul strengthens his point by appealing to the status of the children in such marriages. If it were otherwise, that is, if there were no sanctification of the unbelieving spouse, your children would be unclean; but now, as it stands, because the unbelieving spouse is sanctified, your children are holy. Paul assumes that Christian parents instinctively regard their children as belonging to God, not as "unclean." If the marriage itself were defiling, then the children born from it would also be defiled; but Christians do not and cannot think of their children that way. Therefore, the marriage must be considered holy in a relative sense, and the unbelieving spouse must share in that sanctified status.
From this biblical teaching we derive the principle of the covenant inclusion of infants. The text does not teach infant baptism directly, but it expresses the principle that leads to it: children of believers are to be treated as belonging to the covenant community.
This position is strengthened by the consideration that the "else, now" construction manifests an underlying principle that goes back to the Old Testament. The purpose of a sanctified marriage is to produce godly seed. This is clearly taught in Malachi 2, and must be seen as providing the basic background to Paul's statement.
Mal. 2:15 speaks of God seeking godly seed, or the seed of God. These are offspring who belong to the covenant and are raised in covenant fidelity. Paul's argument in 1 Cor. 7:14 is structurally identical. If the marriage were defiling, the children would be "unclean." But they are holy, having been set apart by virtue of being the offspring of a believing parent. Paul is using the same covenant logic as Malachi: "The covenant holiness of the parent secures the covenant holiness of the children.
1 Cor. 7:14 demonstrates that the federal holiness of children is not confined to the Old Testament but continues into the New Testament; and, as might be expected, is extended rather than restricted. Whereas the temporal and national boundaries of the OT administration meant that unbelievers could defile the marriage and threaten the status of covenant children, as seen in Malachi; the spiritual and heavenly dispensation of the NT now exerts its influence to sanctify even mixed marriages for the sake of the children who are to be brought up and nurtured in the covenant of grace. Where sin abounded grace did much more abound.