Question: Could you please explain what it meant in 1st Corinthians 7:14 by "the unbelieving husband has been sanctified through his wife"?
Response: The issue of divorce when it comes to believers married to non-believers is an emotionally loaded one for two reasons, both of which Paul addresses in this context. On the one hand, it is easy to see how someone who coverts to Christianity and the spouse won't, or finds out after the fact that the spouse is not really a Christian, or realizes too late that the Bible warns against marrying unbelievers for good reason, it is easy to see, I say, how this "problem" of spousal unbelief might lead a faithful follower of Christ to seek dissolution of the marriage for that reason alone. On the other hand, it is also and perhaps equally true that a believer who genuinely loves and cares for an unbelieving spouse, might otherwise feel constrained to try and stay in a "marriage" which has become no marriage in fact (for all the reasons one might imagine) simply out of a desire to try and "save" the unbelieving (and unwilling) spouse. Both of these approaches are incorrect, and Paul explains that in this section of chapter seven. The "bottom line" of 1st Corinthians 7:13 is that unbelief of the partner is not on its own grounds for divorce. That is certainly not to say, however, that there can be no grounds for divorce from an unbeliever (or believer, for that matter), rather that if this unbelief is the only "problem", then it is better to stay married.
As to the next verse, the one about which you are specifically asking, this must be understood in light of the previous verse just discussed, for 1st Corinthians 7:14 is addressed to that perhaps unique situation of "the only problem being unbelief" type of marriage. In verse 14, Paul reassures women in this difficult situation (of being married to a "good" unbeliever) that their lives are not meaningless in the Lord (even if they stay married to this unbeliever), nor are their children somehow tainted because of their spouse's lack of belief - their own faith "sanctifies" the union (marriage is a sacred bond invented by and sanctioned by God for unbelievers as well as believers). This is an important clarification for us to receive, because in many instances in life, it is the bad which permeates/flavors the good (like leaven), but in some instances, it is the good which permeates the bad (as having the preservative property of "salt": Matt.5:13; Mk.9:30; Col.4:6). Paul assures us here that in the case of marriage, the believer is "the salt", so to speak, and makes the entire union, including the offspring from it, "holy" in God's eyes (cf. Lev.2:13). That is to say, along with the believer himself/herself, the marriage, the unbelieving spouse, and their children are all "set apart to God" for the special protection and oversight that falls to the lot of believers. In the same way that the presence of believers in organizations and countries brings blessing and protection from God (especially of believers who walk close to the Lord - compare Lot's deliverance for Abraham's sake), so in marriage too this principle obtains. This does not mean that the unbelieving husband (or the children) are automatically saved, but it does mean that they are treated by God as belonging to the believing spouse, falling under that spouse's "umbrella" of blessing and protection.
For the unbelieving husband is set apart as holy to God on account of [his relationship with] the [believing] wife, and the unbelieving wife is set apart as holy to God on account of [her relationship with] the [believing] brother. For otherwise [hypothetically, i.e., without this principle of sanctification in the believer] your children are unclean, but as it is in reality, they are holy.
1st Corinthians 7:14
The upshot is that finding oneself in a marriage to a person who does not share one's own commitment to Jesus Christ does not in and of itself mean that such a believer can have no fruit for God (on the contrary, the believer is a cause of blessing for his/her unbelieving spouse and family). On the other hand, this principle only applies in cases where the unbeliever is willing to continue in the marriage as an honorable partner in a way which allows the believer to really be a believer. The verse immediately following (7:15) also makes it clear that if the unbelieving husband is not willing to continue in a marriage with a believing wife (in our society where women have a say in their own destinies, this would apply to husbands too), the wife is free from the responsibility of trying to maintain a union under such difficult or even impossible circumstances ("unwillingness" can be demonstrated in a number of ways: e.g., infidelity or abuse, to name but two). In such circumstances, she should feel no guilt about giving up when her husband has already given up (in deed, even if not in word), because, as verse 16 goes on to say, in effect, "you can't tell whether or not [your influence and prayers] will save your husband".
You might also want to have a look at the following links:
Jephthah's Daughter, Marriage, Divorce, and Remarriage.
Marriage of Believers and Unbelievers
Blessing by Association
A Conversation about Divorce and Remarriage
Divorce and Remarriage
More on Divorce and Remarriage
Yours in our Lord Jesus Christ,