Here is an excerpt on this phrase from an online Bible commentary which may be found at:
THE HUSBAND OF BUT ONE WIFE: The phrase has been the center of some disagreement among commentators. It may indicate that there were some Christians as late as 60 AD who had several wives under the permission of Mosaic of Roman laws, and who were allowed to keep them in this early period of transition. It seems almost certain that such brethren had married more than one wife before they accepted the Truth. The fact, however, that such men were baptized and received into fellowship indicates that they were accepted as the Truth found them and were not required to sever any existing marriage ties as a condition of baptism. No restrictions were placed upon such men except that they could not hold the office of a bishop.
Other commentators feel (wrongly, it seems) that this is a command that elders may have only one wife for all time, that is, that they may not remarry if their first wife should die. (JT writes of Tertullian, a bishop of the early apostasy, who flourished about 100 years after the apostle John. Among heresies either introduced or given formal acceptance by this man was the disapproval of such second marriages: Eur 1:437; cp Mosheim's "Ecclesiastical History" 83, 84.) But there is no Scriptural command or precedent for this. There is just as good reason for a widower to marry as for a bachelor to marry (1Co 7:8,9).
The most logical and simplest explanation of this passage is as follows: The Greek phrase is "a man of one woman", or a faithful husband, not guilty of any indiscretion. In the midst of very lax Greek standards of marriage and adultery, a bishop must be very careful to stand apart and to remain faithful to his wife. He must give no appearance (even if innocent) of following the prevailing trends of immorality. If we view this phrase in this light then this phrase is consistent with 1Ti 5:9, where it is said certain women should have been the "wife of one man". (At no time were women permitted to have several husbands. And this could not mean that a woman who had been widowed twice was any less worthy of care simply because of her two marriages. It must mean instead that she should have been wholly faithful to each of her husbands in turn.)
It is also a possibility that Paul has divorce in mind. Divorce was as common in Paul's day as it is today. In this view, a brother who was divorced and remarried, for whatever reason, would be excluded from any 'official' position of service in the church -- although being received into fellowship.
Note the contrast between first century Christianity and the apostasy which was to arise. One had the healthy, God-given attitude that marriage was honourable; the other commanded the unnatural (for most) condition of celibacy to its bishops (1Ti 4:3).
I hope you have found this helpful.